c) Without a loss on the next game by the Vikings, the Packers don't make the playoffs.

Transcription

1 Page Project Due /03/04 Score /48 Name Logic Review: For information on basic logic, review the material in Appendix C: Introduction to Proofs of the text. Additional material can be found at: ( point) In the following compound propositions there are two simple propositions connected by a conditional. Indicate for each pair of simple propositions which proposition is the necessary condition for the other and which proposition is the sufficient condition for the other. a) If I pass my algebra class I'm done taking math. b) Unless the engine is overhauled the car won't run. c) Without gas in the tank the car won't run. d) Henry won't go to school unless he feels better.. (3 points) Each proposition below is followed by a list of statements. Indicate which of the statements are equivalent to the original proposition. a) If the weather doesn't improve the game is cancelled. i) If the weather improves the game won't be cancelled. ii) If the game is cancelled the weather does not improve. iii) If the game is not cancelled the weather does improve. iv) Unless the weather improves the game is cancelled. v) Unless the game is cancelled the weather improves. vi) The weather improves or the game is cancelled. b) Without the proper motivation Sally won't do her homework. i) If Sally has the proper motivation she will do her homework. ii) If Sally hasn't the proper motivation she won't do her homework. iii) If Sally does her homework she has the proper motivation. iv) If Sally doesn't do her homework she does not have the proper motivation. v) Unless Sally has the proper motivation she won't do her homework. vi) Unless Sally does her homework she does not have the proper motivation. c) Without a loss on the next game by the Vikings, the Packers don't make the playoffs. i) If the Vikings have a loss on the next game, the Packers make the playoffs. ii) If the Packers make the playoffs the Vikings have a loss on the next game. iii) If the Packers don't make the playoffs the Vikings do not have a loss on the next game. iv) If the Vikings don't have a loss on the next game, the Packers don't make the playoffs. v) Unless the Packers make the playoffs, the Vikings do not have a loss on the next game. vi) Unless the Vikings have a loss on the next game, the Packers make the playoffs. 6/30/03 Madison Area Technical College

2 Page 3. (3 points) Indicate which of the following statements are tautologies (always true) and explained how you arrived at your answer. When the statement is a tautology then the two propositions separated by the are equivalent. a) ( p q) ( q p) b) ( ~ p ~ q) ( q p) c) ( ~ p ~ q) ( q p) d) q [ q ( q p) ] e) [( p q) ( ~ q ~ p) ] ( q p) f) [( p q) r] [ q ( p r) ] 4. (3 points) Represent the following statements with quantifiers ( ), using the predicates given. In addition for each sentence state what you think is the intended domain of the variables. m(x)"i mean x" ; s(x)"i say x". a) I mean whatever I say. b) I say whatever I mean. c) Unless I mean something, I don't say it. f(x)"x is fun" ; r(x)"x is immoral" ; t(x)"x is fattening" ; g(x)"x is illegal". d) Everything that's fun is illegal or immoral or fattening. p(x)"x is a politician" ; h(x)"x is honest". e) Not every politician is dishonest. f) There are honest politicians. 6/30/03 Madison Area Technical College

3 Page 3 5. ( point) State the negation of each of the following by changing universal quantifiers to existential quantifiers and vice versa. a) I mean whatever I say. b) There is at least one honest man. c) Everything that's fun is illegal or immoral or fattening. d) For any real number x, f (x) > 0. e) ε {( ε > 0) δ ( x [( 0 < x a < δ ) f ( x) f ( a) < ε ])} 6. ( point) Classify each of the following arguments as either valid or invalid. Justify your choice. a) Premise: If r is a prime number, then r is a prime number. Premise: q is a prime number. Conclusion: q is a prime number. b) Premise: Unless you pay your rent you will be evicted from your apartment. Premise: If you are evicted from your apartment, you will move back in with your parents. Premise: You move back in with your parents. Conclusion: You did not pay your rent. 7. ( point) Draw a valid conclusion from the stated premises and detail your reasoning. Premise: If there is at least one person who fails to lose weight on diet A, then all of the brochures must be revised and the advertising manager will be fired. Premise: Some of the brochures were not revised. Conclusion: 8. ( point) If r is a prime number, then r is a prime number. Is answer a prime number? Justify your 6/30/03 Madison Area Technical College

4 Page 4 9. ( points) Justify each of the following set operation identities. a) A B A B b) A B A B c) A ( B C) ( A B) ( A C) d) A ( B C) ( A B) ( A C) 0. ( points) Prove the following: a) The sum of the first n natural numbers is given by n( n + ). 6/30/03 Madison Area Technical College

5 Page 5 n( n+ ) b) The sum of the cubes of the first n natural numbers is given by. What if anything is wrong with the following demonstration by Mathematical Induction that if one rabbit is white all rabbits are white? Hypothesis: In any set of n rabbits, if one of the rabbits is white, all of the rabbits are white. Base Case: Induction Step: Suppose we have a set with one rabbit. If this rabbit is white, then all of the rabbits in this set are white. So the result is true for n. Consider a set with n + rabbits. Suppose one of the rabbits is white. Remove another rabbit (call it "Bunny") from the set. Then we are left with a set of n rabbits, one of which is white. By the Induction Hypothesis, all of these n rabbits are therefore white. Take one of these white rabbits from the set and replace it with Bunny. Again we have a set of n rabbits. Since we know that all of the rabbits in this set, with the possible exception of Bunny, are white, we have a set of n rabbits with one of the rabbits being white. So by the Induction Hypothesis, all of these n rabbits, including Bunny, are white. Hence, if one of the rabbits is white, all n + rabbits are white. Suppose that it were true that in any set of two electrons, if one electron has charge e so does the other. Would it now be true that in any set of n electrons, if one of the electrons has charge e, all of the electrons have charge e? Explain your reasoning 6/30/03 Madison Area Technical College

6 Page 6. ( points) A function that maps input from a set A to output in a set B is called a bijection if and only if the function is both one to one and onto. Here onto means that the range of the function is precisely the set B. a) Consider a one to one function from the set {,, 3,, n} to{,, 3,, n}. Must this function be a bijection? Explain your answer. b) Consider an onto function from the set {,, 3,, n} to{,, 3,, n}. Must this function be a bijection? Explain your answer. c) The set of all bijections from the set {,, 3,, n} to{,, 3,, n} is called the Permutation Group of degree n. How many elements (this is called the order of the group) does the Permutation Group of degree n have? Matrix Operations: 3. ( point) Given that Compute the following: AB 3C T 3 5 A , B and C 7 3. T AA T A A 8//03 Madison Area Technical College

7 Page T 4. ( point) Given that 3A B 5 T and A + 5B 3 4, 6 solve for the matrices A and B. A B 5. ( point) Prove that the product of two symmetric matrices is symmetric if and only if the matrix product commutes. 6. ( points) If A is an m x n matrix and B is an n x k matrix, prove that a) Every column of AB is a linear combination of the columns of A b) Every row of AB is a linear combination of the rows of B 6/30/03 Madison Area Technical College

8 Page 8 7. ( point) Prove that for two n x n matrices A and B that the trace( AB) trace( BA). Systems of Equations: 8. (3 points) Use the Gauss-Jordan elimination method to find all possible solutions of the following linear systems. Indicate the elementary row operations you used. a) x + y z 3 3x y + 3z 5 5x + 3y + z b) x + 3y z 8 x + y + 3z 3 4x 3y + 5z 0 x y z x y z 6/30/03 Madison Area Technical College

9 Page 9 c) x + y 3z + w x 3y + z + 3w 3x y z + 4w x 4y + 5z + w 3 x y z w 9. ( point) Go to the Linear Algebra Toolkit site Transforming a matrix to reduced row echelon form where you can enter an m x n matrix and the program will display a sequence of row operations that reduces the matrix to reduced row echelon form. Use this program to solve the following linear system. x + x x3 + x4 x + 3x + x3 3x4 0 x x + x3 6x4 x + 6x3 3x4 7 From the results state the complete solution x x x3 x4 6/30/03 Madison Area Technical College

10 Page 0 0. ( point) Go to the Linear Algebra Toolkit site Calculating the inverse using row operations where you can enter a square matrix. If the n x n matrix is nonsingular, the program calculates the inverse. Use this program to find Click on the link that reads can be transformed by a sequence of elementary row operations to show the steps. Now use the inverse to solve the following systems: x + y 3z 8 x + 5y + 6z 35 7x 3y + 4z 4 x y z x + y 3z a x + 5y + 6z b 7x 3y + 4z a+ b x y z. ( point) The Linear Algebra Toolkit uses exact integer arithmetic to perform calculations, so only integer or fraction input is allowed. This has disadvantages if the matrix has its origins in data where a decimal representation is more typical. The Matrix Algebra Tool V. at the site uses an interface similar to Matlab and allows decimal input. Use the Example button to examine input syntax. Note that matrices are input in the white space above the Formulas: The calculations you wish to perform on the matrices are entered in the Formulas line. When you press the Compute button the results of the calculations are displayed in bottom white space. Use this program to solve the following system of equations:.5x 0.387y +.54z x y +.7z x.9y z From the results state the complete solution x y z 6/30/03 Madison Area Technical College

11 Page. (3 points) Solve each of the following three equations for z. x y z x + y + z x y + z 8 a) In Winplot open the 3-dim window. For each formula for z above enter an Explicit function in the Equa/Explicit menu. From View pick Axes to get a better 3D perspective of the surfaces. Be sure to use a different color for each surface. This is chosen under color of the Equa/Explicit menu. Solve this system of equations and enter the coordinates of the solution using the Equ/Point/Cartesian menu. Be sure the color of this point is distinct from the color of the three surfaces. Use PgUp (or the View Menu) to "zoom in" and the left and right arrow keys to rotate the graph until you get a "good view" which clearly shows the three surfaces and the solution point. Print and attach your graph. What kind of geometric surfaces does each equation describe? What is the significance of the solution point with respect to the three surfaces? b) Now solve each of the following three equations for z. x y z x + y + z 3x + 3y + z For each formula for z above enter an Explicit function in the Equa/Explicit menu. From View pick Axes to get a better 3D perspective of the surfaces. Be sure to use a different color for each surface. Show that this system is consistent with infinitely many solutions along a line. Determine the form of this solution. In this form let z t where t is any real number. Solve for x and y and in terms of t. In Winplot use the Equa/Curve menu to enter the form of the infinitely many solutions. Set t lo -5and t hi 5. Set the pen width to and choose a dominant color to make this curve more prominent. Use PgUp (or the View Menu) to "zoom in" and the left and right arrow keys rotate the graph until you get a "good view" which clearly shows the three surfaces and the line of solutions. Print and attach your graph. What kind of geometric surfaces does each equation describe? What is the significance of the line of solutions with respect to the three surfaces? 6/30/03 Madison Area Technical College

12 Page 3. ( points) a) Determine by elementary row operations. Indicate the elementary row operations you used. b) Now use the inverse to solve the following system: x + y z a+ b 3x 5y + z 3a b x 4y + 3z a b x y z 4. (4 points) Given that A A 3 and B B compute the following: AB A B B A BA B A A B ( ) BA ( ) AB 6/30/03 Madison Area Technical College

13 Page 3 5. ( points) Prove if A is a nonsingular symmetric matrix that A is also symmetric. 6. (3 points) Prove that if A and B are n x n matrices with AB In then BA In. 7. ( points) k T k a) Prove that for any n x n matrix A and any whole number k ( A ) ( A ) T. 6/30/03 Madison Area Technical College

14 Page 4 b) Prove that for any n x n nonsingular matrix A and any whole number k ( ) k k ( ) A A. 8. (3 point bonus) In each case below display a x matrix which transforms the figure on the left into the figure on the right. a) 6/30/03 Madison Area Technical College

15 Page 5 b) c) Calculate the correlation coefficient for the following set of 3 data points: i x i y i /30/03 Madison Area Technical College

16 Page 6 Group Project Due //04 Score / Name Name Name Name Do one of the following three problems for points. You may do extra problems to earn up to a total of 4 extra credit points.. Newton s method in multiple dimensions. The Background For functions of a single variable the equation f ( x ) 0 can be solved approximately by an iteration known as the Newton Raphson method. The method can be derived from the linear or tangent approximation to f ( x ). Suppose we have an initial guess x n to a solution or root of f ( x ) 0. The linear approximation about x xn is f ( x) f ( x ' n) + f ( xn)( x xn) if we set f ( x ) 0 this gives an equation for the change f ( xn ) f ( xn ) Δ x x xn which gives the recursion x f ' n+ xn. If the method works the relative error of the ( x ' n ) f ( xn ) Δx approximation becomes very small as we iterate. Whether in fact this happens usually depends on the initial xn guess x 0. If the derivative vanishes at the desired root, the method can be very inefficient or actually divergent. Now consider the system of m equations in m variables ( ) notation x x, x,, xm with vector length f x, x,, x 0for j m. Adopting the vector 6/30/03 Madison Area Technical College j approximation to equation j becomes f ( x) f ( x ) + x ( x ) m m x x, x,, xm x, x,, xm xj, the linear j m f j j j n ( i n i ), where the partial derivatives are x i i n n n n F x f j m n by Fji, where again the partial xi x n x n, x,, n x n and the m x column vector Δ x ( ) m i xi xn. The linear i evaluated at x ( x ),( x ),,( x ). Define the m x m matrix ( ) derivatives are evaluated at ( ) ( ) ( ) m f j x fj xn + FjiΔxi i approximation to equation j can now be written as the j th row of a matrix product ( ) ( ) Since we want fj ( x ) 0 given by bj fj( xn) to the recursion ( ),( ),,( ), this yields the system of linear equations FΔ x b, where the m x column vector b is. If the Jacobian matrix F is nonsingular then there is a unique solution Δ x F b. This leads xn+ xn +Δ x xn +Δ x xn +Δx m m As in dimension we iterate until the relative error Δx given by the ratio becomes very small and convergence of the method depends on the initial guess xn x0 ( x0), ( x0),, ( x0). m.

17 Page 7 The Problem The system of equations ( ) 36ysin( π x) 6x + 45y x y 9 e 5 6y 6( e ) has three solutions: one at the origin, one in the third quadrant and one in the first quadrant. Using Newton s method, approximate the solution in the first quadrant to a 6 relative error smaller than0. Depending on your initial guess you may need more or less than 6 iterations. Record your results in the following table. Programming the method into Excel or a graphing calculator will make this calculation manageable. As you approach the solution the quantity f ( x, y) + f ( x, y) should approach zero. n x n y n Δ x Δ y Δ x +Δy x n + yn ( ) ( ) f x, y + f x, y. The Least Squares solution to over determined linear systems. The Background Consider the system of m linear equation in n unknowns with m > n. In matrix form this is expressed as Ax bwhere A is an m x n matrix, x is an n x column vector and b is an m x column vector. We will assume that A has column rank of n. This means that in reduced row echelon form A has n nonzero rows followed by m-n rows of zeros. In general for an arbitrary m x column vector b the system Ax b would be inconsistent, i.e., the reduced row echelon form of the augmented matrix [A b] would not have all zeros in the bottom m-n rows of the last column. Thus, for most choices of b there is no x that solves the system. However, there is an x that minimizes the distance between Ax and b. This is called the least squares solution of the system. The squared distance from b of Ax is a function of the n coordinates in x and is given by the expression m m n m n n n f ( x) b Ax ( bj ( Ax) ) ( ) j bj Ajixi bj bj Ajixi + Ajk xk Ajixi j j i j i i k m n n n m m n n n T T T T b b bjajixi + AkjA ji xixk b b bjajixi + ( A A) xixk ki j i i k j j i i k For fixed b the distance from Ax to b has no upper bound, but it must have a minimum since it is bounded below by zero. The partial derivatives of f with respect to each coordinate of x must vanish at the value of x that minimizes ( ) x f x. Now k x j δ and kj T A A is a symmetric matrix so that 6/30/03 Madison Area Technical College

18 Page 8 T ( ) m n n n f x i xi x k ba j ji + AA xk + xi xp x ki j i p x i k p x p m n n n m n n T T T ba j jiδip + ( AA) ( δipxk + xiδkp ) ba j jp + ( AA) xk + ( AA) xi ki kp pi j i i k j k i m n n T T T T T Apjbj ( A A) xk ( A A) xk ( A A x Ab) + + ( ) pk pk p j k k f At the minimizing value of x, for each p between and n 0. Thus the minimizing value of x must satisfy the xp 0 T T equation ( A A) x A b where the column vector of zeros is n x. Now, if the row rank of A is n, A is row 0 equivalent to an RREF matrix which in partitioned form has I n above m n rows of zeros. To be more precise there is In an elementary m x m matrix E such that EA. 0 m n, n x b x b Thus, the equation Ax A b is equivalent to xn bm x x b In Inx x x x b EAx EA E and if b 0 this leads to 0 m n, n 0m n, nx 0 m n, x n x n bm x 0 x x 0 xn 0 x 0 EAx. So the equation Ax 0 has only the trivial solution, x m n, xn x x m m m For any x consider ( ) ( ) ( ) ( ) ( ) (( ) ) T T T T x A Ax Ax Ax Ax Ax, i i, Ax Ax i, i, Ax. i, i i i xn 6/30/03 Madison Area Technical College

19 Page 9 If T A Ax is the n x zero vector, then 0 ( ) m T T x A Ax so ( Ax) i, i zero if each term is zero. Thus, each ( Ax ), 0 or Ax 0. But 0 i implies that x is the trivial solution. Therefore T equation ( ) The Problem A A x T T A bgiven by ( ) 0. Now a sum of real squares is only equal to Ax has only the trivial solution, so A Ax 0 also T A A is nonsingular. Thus, there is a unique least squares solution to the T x A A A b. a) Show that if m n that the least squares solution actually solves Ax b. T b) Find the solution in the least squares sense of the following problem: x y x ˆ y β 0 x3 y3. ˆ β x m y m ˆ 0 β ˆ β 6/30/03 Madison Area Technical College

20 Page 0 3. Solving tridiagonal linear systems (Thomas Algorithm) The Background A square matrix A is called tridiagonal if all of the elements in positions more than one diagonal removed from the ai if i j 0 ci if i j main diagonal are zero. More precisely Aij,. A schematic of such a tridiagonal matrix is di if i j 0 if i j > a d c a d c3 a3 d3 0 A. Now the linear system Ax b for an n x column vector b can be solved cn an d n cn an using Gaussian elimination. A much faster procedure however is to use a recurrence. We will assume that none of the b a s are zero. From the first row dx d x. Assume that we can find γi and βiwith x i i βi xi+. This leads a γi to a recurrence for γ i in terms of ai, ci, di and γi and a recurrence for β i in terms of bi, ci, γi and βi with the b initial conditionsγ a and β. Thus, from A and b values can be obtained forγi and βiwith i n. a d Furthermore, it s not difficult to show that xn βn. The backwards recurrence x i i βi xi+ then calculates the γi solution for i running from to n. The Problem Solve for the recurrence relations of γi and β i. recurrence relation of γ i : recurrence relation of β i : 6/30/03 Madison Area Technical College

21 Page Apply this method to solve the following system of equations x x x x x x x x x x x x x Use Excel or a graphing calculator to complete the entries in the following table. State decimal solutions to the nearest ten thousandth. i a i c i d i b i γ i β i x i 6/30/03 Madison Area Technical College

22 Page Project Due /7/04 Score /48 Name. ( points) Determine the number of inversions and the parity (even or odd) of the following permutations of,,3,4,5. { } a),, 3, 4, 5 b) 5, 4, 3,, c) 3, 4,, 5,. (3 points) Go to the Linear Algebra Toolkit site Calculating the determinant using row operations where you can enter an n x n matrix and the program will display a sequence of row operations that allows for a calculation of the determinant. Use this program to calculate the following determinants (0 points) Calculate the following determinants: a) /30/03 Madison Area Technical College

23 Page 3 b) c) a b e q a b f r a b g s a b h t d) 3 a b c 0 d e f e) a b 0 c 0 b a b 0 a b b a b c c f) α γ β β α γ γ β α 6/30/03 Madison Area Technical College

24 Page 4 g) α δ γ β β α δ γ γ β α δ δ γ β α 4. ( points) Solve the following equations for λ. 0 a) det λi b) det λi /30/03 Madison Area Technical College

25 Page 5 5. ( point) Prove for real a, b and c that a 0 a a b a c b 0 c a b b b a b c 6. (4 points) A is an n x n matrix with rows R, R, Rn, Rnand columns C, C, Cn, Cn. That is, R R A [ C, C, Cn, Cn], with Ri Cj j. A cyclic permutation that shifts the rows of A up by one i Rn R n R R 3 position results in the matrix B, while a cyclic permutation that shifts the rows of A down by one position Rn R Rn R results in the matrix D. Similarly matrices E and F are generated by cyclic permutations of the columns of Rn R n A one position to the right and left respectively, E [ Cn, C, Cn, Cn ], F [ C, C3, Cn, C]. a) Express det[ B] R R 3 det Rn R det A. in terms of the [ ] 6/30/03 Madison Area Technical College

26 Page 6 b) Express det[ D] Rn R det Rn R n in terms of the [ ] det A. c) Express det[ E] det [ C, C, C, C ] in terms of the [ ] n n n det A. d) Express det[ F ] det [ C, C, C, C ] in terms of the [ ] 3 n det A. e) For 3-D vectors α αx, αy, αz, β βx, βy, βz and γ γx, γy, γz the dot product γx γy γz γ α β det αx αy αz. From a) what can you deduce about β γ α and α β γ? βx βy βz 7. (4 points) Solve the follow system of equations using Cramer s Rule. You must show your set up. a) 3x y 4 7x + 5y 6 x y b) 3x + y z 4 4x 3y + z 6 x + 5y + 3z 9 x y z 6/30/03 Madison Area Technical College

27 Page 7 ax + by + az a c) bx + by + bz b cx + by + bz c x y z What condition must be satisfied by a, b, and c so that your stated solution is valid? What condition must be satisfied by a, b, and c so that your stated solution is unique? 8. ( points) Determinants can also be computed at the Wolfram Alpha website: To calculate the determinant of an n x n matrix A enter the following: det( (row of A), (row of A), (row n of A ) ). The syntax requires that rows be stated in parentheses with the row elements separated by commas. This is illustrated in the example below. Note that the matrix entries are not restricted to numbers. 6/30/03 Madison Area Technical College

28 Page 8 Use Wolfram Alpha to solve the following system of equations by using Cramer s Rule and state the solution below. 0.5x 0.69y +.543z x y 0.05z x.793y z x y z 6/30/03 Madison Area Technical College

29 Page 9 9. (3 points) For each matrix A below, display the adj( A ). a) A adj ( ) A b) A adj ( ) A c) 4 A 3 3 adj ( ) A 0. (3 points) Given that A is an n x n matrix, simplify the following: a) A adj( A) b) adj( A) A c) det adj( A) Suppose now that A is nonsingular, express the following in terms of A d) adj( A ) T e) adj( ) A Suppose that A is nonsingular, express the following in terms of A f) ( ) adj A g) adj adj( A) 6/30/03 Madison Area Technical College

30 Page 30 T. ( points) A nonsingular matrix A with the property that A A is said to be orthogonal. What values can the determinant of an orthogonal matrix have? Support your answer with an argument.. ( points) If A and B are n x n matrices and B is nonsingular prove that det ( AB ) ( A) ( B) det. det 3. ( points) Prove that an n x n matrix A is nonsingular if and only if the adj( A) is nonsingular. 4. (4 points) a) If A is an n x n matrix prove that f ( λ) det ( λi A) n is a polynomial of degree n with a leading coefficient of. b) Prove that λ 0 is a root of f ( λ ) if and only if A is singular. 6/30/03 Madison Area Technical College

31 Page 3 5. (4 points) a) Prove that for any symmetric matrix A that the adj( A) is also symmetric. b) Prove that if A is nonsingular and the adj( A) is symmetric then A must be symmetric. c) (4 point bonus) If A is singular and the adj( A) is symmetric, must A also be symmetric? Support your answer with an argument. 6/30/03 Madison Area Technical College

32 Page 3 Group Project Due /5/04 Score / Name Name Name Name Choose two of the following five problems. Each problem is worth 6 points. You may do extra problems to earn up to a total of 8 extra credit points. x y. a) Expand the equation x y 0. In the x y plane what figure does this generate? x y b) Three points in the plane not all on the same straight line determine a circle. A simple geometric construction of the circle proceeds as follows. Construct two line segments between different pairs of the three points. Each of these segments is a chord of the circle. Construct and extend the perpendicular bisectors of each chord. These meet at the center of the circle. The segment from this point of intersection to any of three original points is a radius of the circle. Despite the ease and simplicity of this construction, at times (such as in computer graphics or surveying) it would be beneficial to have an analytic representation of the circle determined by the three points. This can proceed according to the same steps as in the geometric construction. However, the algebra, while not difficult is messy and seems to hide the solution. A compact form for the equation of the circle would seem to be in order. Calling the three points ( x, y ), ( x, y ) and ( x3, y 3) give an argument that proves that the equation of the circle through these three points is given by x + y x y x + y x y 0. x + y x y x3 + y3 x3 y3 6/30/03 Madison Area Technical College

33 Page 33 Given the points ( 0., 7.4), ( 6.8,.6) and ( 3, 4.) determine the equation of the circle through these three points. In Winplot -dim plot these three points using Equa/Point. Then in Equa/Implicit enter the equation of the circle. Print out and hand in your graph. In three dimensions four points not all in the same plane lie on the surface of a sphere. Calling the four points ( x, y, z ), ( x, y, z ), ( x3, y3, z 3) and ( x4, y4, z4) write an equation for this sphere. In n dimensions n + points not all in the same n-dimensional hyper-plane lie on the surface of an n-dimensional hyper-sphere. Calling the n + points ( x, x, x3,, xn ), ( x, x, x3,, xn ), and ( xn+,, xn+,, xn+,3,, xn+, n) write an equation for this hyper-sphere. 6/30/03 Madison Area Technical College

34 Page 34. Vandermonde determinants. Expand the following determinants and simplify your answers. x x x x x x x3 x3 n n n n 3 n 3 n 3 x x x x x x x x x n n n n n x x x 6/30/03 Madison Area Technical College

35 Page Orthogonal matrices An n x n matrix A is called orthogonal if the dot products of its row vectors are zero for different rows and for the n n 0 if i k same row, i.e., if A ri is the i th row of A, then Aij ( Ari) and ( A ) ( ) j ri Ark A j j i jak j δ ik if i k j j We say that the rows of A are orthonormal. a) If A is an orthogonal matrix calculate T AA and A T A.. b) Are the columns of an orthogonal matrix orthonormal? Justify your answer. x x c) An n-dimensional column vector x of real numbers has a length given by the Pythagorean Theorem as xn x n x j j. If A is an orthogonal matrix what is the length of Ax? d) Consider A( ) ( ϕ ) ( θ) ( θ) ( ϕ ) ( θ ) ( ) ( ) ( ) ( ) ( ) sin ( ϕ) 0 cos( ϕ) cos cos sin sin cos ϕθ, cos ϕ sin θ cos θ sin ϕ sin θ. Show that A( ϕ, θ ) is an orthogonal matrix. e) Compute det A( ϕ, θ ). 6/30/03 Madison Area Technical College

36 Page 36 f) The standard unit vectors i, ĵ and ˆk of three dimensional space can be represented as the following column vectors: 0 0 i 0, ˆj and k ˆ 0. For 0 ϕ π and 0 θ < π define the new vectors i A( ϕθ, ) 0, ˆ j A( ϕθ, ) and kˆ A( ϕθ, ) 0. Compute the following: 0 i i i ˆ j i kˆ ˆj ˆ j ˆ j kˆ kˆ kˆ i ˆ j ˆ j kˆ kˆ iˆ g) In Winplot 3-dim graph using different colors the vectors i, ĵ and ˆk. Use A for ϕ and B for θ and generate the graph with Equa/segment choosing a b c 0 and d, e and f defined as the components of the vector. Put an arrow at p. In the Anim/Parameters A-W menu set the lowest value of A (set L) to 0 and highest value of A (set R) to PI. Similarly set the lowest value of B to 0 and the highest to *PI. Describe what happens to your graph as you animate on A and B. Print out and attach such a graph with A and B at intermediate values in their domain. 0 0 h) Consider B 0 0. Show that B is an orthogonal matrix. 0 0 i) Compute det[ B ]. j) What is true about the determinant of an orthogonal matrix? k) Does applying the matrix B to the standard unit vectors i, ĵ and ˆk preserve the right handedness of the unit vectors? Explain your answer. 6/30/03 Madison Area Technical College

37 Page Permutation matrices a) Consider the permutation from {,, 3 4} onto {,, 3, 4} given by p() 3, p( ), p( 3) 4, p( 4). What is p? To answer this question is fill in the following: p ( ), p ( ), p ( 3 ), p ( 4 ) b) Let ê j, n designate the j th row of the n x n identity matrix. Let P be the 4 x 4 matrix whose j th row is ê p ( j ),4. For the permutation stated in part a), P. If x x x x3 x4 and A fill in the following: Px PA AP c) Let B be the 4 x 4 matrix whose j th row is ê p ( j ),4. Fill in the following: B PB BP How B related to P?. d) Compute det ( P ). det( P ) e) Let p be a permutation from {,, n} onto {,, n}. Let P be the n x n matrix whose j th row is ê p ( j ), n. If x x x what is the effect of multiplying x by P? xn 6/30/03 Madison Area Technical College

38 Page 38 If A is an n x n matrix what is the effect of multiplying A by P (i.e., PA )? What is the effect of multiplying P by A (i.e., AP )? f) What is P? g) In general what can be said about the value of det ( P )? 5. Consider the n by n matrix, G(n), defined by the following formula: Gn ( ) aδ + bδ + cδ i, j n i,j i,j i +,j i,j, i.e., ( ) Gn a b c a b c a b c a a b c a b c a a) Determine a recursion for the det G( n). 6/30/03 Madison Area Technical College

39 Page 39 b) Use your recursion to compute det then check your answer with the Linear Algebra Toolkit or Wolfram Alpha. 6/30/03 Madison Area Technical College

40 Page 40 Project 3 Due 3//04 Score /48 Name. ( point) a a a a) Using the vector notation that b a, b, c aiˆ + bj ˆ + ckˆ calculate 3 b 5 a b c c b+ c Solve the following for C and C. C C b) C 6 C c). (4 points) Which of the following are vector spaces? Justify each answer. x a) The set y x R, y R, and z R under vector addition and scalar multiplication. z x b) The set 0 x R and z R z under vector addition and scalar multiplication. x c) The set 0 x> 0, and z< 0 under vector addition and scalar multiplication. z 3/5/04 Madison Area Technical College

41 Page 4 { f c( c R x( x R f x c) )} d) The set and ( ), i.e. the set of functions of a real variable equal to a real constant, under ordinary addition and multiplication. { ( and ( 0 ))} e) The set f f ( x) x x R f ( x), i.e. the set of differentiable functions of a real variable with a derivative equal to zero, under ordinary addition and multiplication. { ( and ( ))} f) The set f f ( x) x x R f ( x), i.e. the set of differentiable functions of a real variable with a derivative equal to one, under ordinary addition and multiplication. g) The set of real m x n matrices with all positive entries under matrix addition and scalar multiplication. 6/30/03 Madison Area Technical College

42 Page 4 3. ( points) Which of the following are subspaces of C (, )? Justify each answer. a) The set of differentiable functions b) The set of continuous functions with no real roots. c) The set of solutions to the differential equation f ( x) f ( x). 4. (3 points) Which of the following are subspaces of M nn under matrix addition and scalar multiplication? Justify each answer. a) The set of n x n matrices with zero trace. b) The set of nonsingular n x n matrices. c) The set of singular n x n matrices. d) The set of diagonal n x n matrices. e) The set of symmetric n x n matrices. f) The set of skew symmetric n x n matrices. 6/30/03 Madison Area Technical College

43 Page Uniqueness of the additive inverse ( point) Prove that in any vector space V a+ u 0 V if and only if a u. { } 6. ( point) Prove that y y C (, ), a R, b R and y + ay + by 0 is a subspace of (, ) C. 7. (3 points) Go to the Linear Algebra Toolkit site Linear independence and dependence where you can enter n vectors from a choice of vector spaces and the program will display a sequence of row operations that determines if the associated homogeneous system has non trivial solutions. Use this program to determine if the following sets are linearly independent. State the results below a) n 3, V R 4, S,, b) n 4, V M 3, S,,, { 3 4 } c) n 4, V P 4, S p () t, p () t, p () t, p () t where () 3 p t + 6t+ t 6t () 3 4 p t 8+ 3t 4t + 9t + 3t () 3 4 p3 t 5 4t+ 8t + 3t + t () 3 4 p4 t 6 + t+ 3t + 9t + 7t 6/30/03 Madison Area Technical College

44 Page (3 points) Go to the Linear Algebra Toolkit site Determining if the set spans the space where you can enter n vectors from a choice of vector spaces and the program will display a sequence of row operations that determines if the associated non-homogeneous system has solutions. Use this program to determine if the following sets span the stated space. State the results below. a) n 3, V R 3, S 5, 5, b) n 5, V M, S,,,, { } c) n 5, V P 4, S p () t, p () t, p () t, p () t, p () t where () () () () () 3 4 p t + 3t+ 5t + 7t p t 3t 5t p t 4+ 6t 3t 3 3 p t 9+ 5t+ 5t 6t p t 8+ 9t+ 0t 3 9. (3 points) Go to the Linear Algebra Toolkit site Finding a basis of the space spanned by the set where you can enter n vectors from a choice of vector spaces and the program will display a sequence of row operations that will construct a basis for the span of these n vectors Use this program to determine a basis for each of the spans in problem 8. State the basis below. 6/30/03 Madison Area Technical College

45 Page (3 points) Go to the Linear Algebra Toolkit site Finding a basis of the null space of a matrix where you can enter an m x n matrix and the program will display a sequence of row operations that will construct a basis for the null space or kernel of the matrix. Use this program to determine a basis for each of the null spaces of each of the following matrices. State the basis below. a) b) c) A A A (4 points) Which of the following sets are linearly independent. Justify your answer. a) In R 4, ,,,, b) In R 4, 4 7 3,,, c) In R 4, ,,, T T T T 6/30/03 Madison Area Technical College

46 Page 46 x x C, {, } d) In (, ) e e. { } e) In C (, ), x x e, e,sinh( x) { } f) In C (, ) cos( x),4,sin ( x ) { } g) In C (, ), cos( x),4,sin ( x ) h) In M 3, , 6 3, 3 5, i) In M 33, , 0, 0 0 0, /30/03 Madison Area Technical College

47 Page 47. (3 points) Find a basis for the span of the following sets. 3 4 a) In R 4,,,, b) In P, { t, t,3} c) In M, 0 0,, (3 points) Determine a basis for the null space or kernel of the following matrices. a) b) c) /30/03 Madison Area Technical College

48 Page ( point) A is a 5 x 9 real matrix. a) What is the maximum dimension of the column space of A? b) What is the minimum dimension of the null space or kernel of A? 5. (5 points) In P given the ordered basis S { t +, t, t}, the ordered basis T { t t, t t, t } natural ordered basis N { t,, t }, a) Determine the transition matrix from N to S , and the b) What is the representation of at + bt + c in the S basis? c) Determine the transition matrix from N to T. d) What is the representation of at + bt + c in the T basis? e) Determine PS T, the transition matrix from T to S. f) Determine QT S, the transition matrix from S to T. g) What is the matrix product ( Q )( P ) T S S T? h) What is the matrix product ( P )( Q ) S T T S? 6/30/03 Madison Area Technical College

49 Page 49 c a i) If [ v] T b, determine [ v ] S. j) If [ ] S a v b c, determine [ v ] T. 6. ( point) Given any two non zero vectors v and and only v tvfor some scalar t. v in a vector space V, prove that { v, v } is linearly dependent if 7. ( point) Prove that the coordinate representation of a vector v in a vector space V with respect to the ordered basis { u, u, un} is unique. 8. ( point) Prove that if S { v v v } is a basis of a vector space V and T { w w w },, n independent subset of V then k n.,, k is a linearly 9. ( point) Prove that a subset of a finite dimensional vector space V is a basis if and only if it is a maximal independent subset of V. 6/30/03 Madison Area Technical College

50 Page ( point) Prove that if V is a vector space V of dimension n, then any subset set of V containing more than n vectors is linearly dependent.. ( point) Prove that an isomorphism between vector spaces is an equivalence relation.. ( points) If LV : W is an isomorphism from vector space V to vector space W, prove that S { v, v,, vn} is T L S L v, L v,, L vn is a basis of W. { } a basis of V if and only if the set ( ) ( ) ( ) ( ) 6/30/03 Madison Area Technical College

51 Page 5 Group Project 3 Due 3/8/04 Score / Name Name Name Name. (4 points) If a function with derivatives to all orders is equal to its infinite Taylor series everywhere on an interval [a, b] the function is called analytic on the interval. Show that the set of analytic functions on an interval [a, b] forms a vector space (of infinite dimension). The set of polynomials {, t, t, } forms a basis for this vector space. Explain. Choose one of the following two problems. Each problem is worth 8 points. You may do an extra problem to earn up to 8 extra credit points.. The set of n x n matrices forms a vector space M nn. a) What is the dimension of M nn? b) Let T nn be the set of n x n matrices whose diagonal elements sum to zero, i.e., T nn is the set set of n x n matrices with zero trace. Show that T nn is a subspace of M nn. 6/30/03 Madison Area Technical College

52 Page 5 c) What is the dimension of T nn? d) Consider the following two sets S and S in T 33 : S S , 0 0, , 0 0, Only one of these sets is linearly independent. Which one is it? Prove your answer. e) Find a basis for T 33 which has the linearly independent set of part d) as a subset. 6/30/03 Madison Area Technical College

53 Page An n x n matrix A is said to be a nilpotent matrix of index k if A k 0. Trivially every zero matrix is nilpotent of k k index. To get nontrivial examples we require that for an integer k >, A 0 but A 0. For example the x 0 matrix A is a nilpotent matrix of index. 0 0 a) Find a 3 x 3 nilpotent matrix of index. b) Find a 3 x 3 nilpotent matrix of index 3. c) Show that every nilpotent matrix is singular. d) Show that if for some nonzero n x column vector x, if Ax λx for a nilpotent matrix A and some scalar λ then λ 0. e) Show that if λ 0 then the matrix A+ λi is nonsingular. f) Show that if A is an n x n nilpotent matrix of index k, then k n. [Hint: There must exist a u vector in R n with k j k A u 0. Consider the span { uau,,, Au, A u} where j k.] 6/30/03 Madison Area Technical College

54 Page 54 Project 4 Due 4/0/04 Score /48 Name. ( point) Given the following pairs of vectors, use the standard inner product to compute ( ) value of the smallest angle between the two vectors. 3 0 a) u v 0 u, v, u, v, and the b) 3 3 u v ( point) Determine the equation of the plane in R 3 that contains ( 4,,3) and is perpendicular to the vector,,. 6/30/03 Madison Area Technical College

55 Page ( points) Given u u, u, u3, v v, v, v3, w w, w, w3 w u v a) Compute ( ) b) If w ( u v) 0 what conclusions can you make concerning w? T 4. ( points) Prove that ( AB, ) tr( A B) defines a valid inner product on the vector space of real n x n matrices. 5. (5 points) a) Prove that ( uv, ) u( tvt ) ( ) dtdefines a valid inner product on [,] C. 6/30/03 Madison Area Technical College

56 Page 56 Given the following pairs of vectors, use this inner product to compute u, v,( u, v), and the value of the smallest angle between the two vectors. b) u() t v() t t, c) u() t, v() t cos( πt) d) u() t t, v() t cos( πt) g x dx x g x dx g x dx. e) If g(x) is continuous on [-, ], prove that ( ) cos ( π ) ( ) ( ) 6/30/03 Madison Area Technical College

57 Page ( points) Using the standard inner product n minimum values possible for u j. j u u u un has a length of in n R. Determine the maximum and 7. ( points) Given that V is an n dimensional vector space with an ordered basis S and that A is an n x n positive T definite matrix, prove that ( uv, ) [ u] Av [ ] defines an inner product on V. S S 8. ( points) Prove that if A is a square matrix, T A Ais positive definite if and only if A is non-singular. 6/30/03 Madison Area Technical College

58 Page (3 points) a) Prove that if A is a positive definite matrix, then all of the diagonal elements of A are positive. b) Explain whether the converse of a) is true. c) Prove that the sum of two n x n positive definite matrices is a positive definite matrix. 0. ( points) a b e f a) Prove that, ae + bf + cg + dh c d g h defines an inner product on M restricted to real entries. 6/30/03 Madison Area Technical College

59 Page 59 b) Using the inner product of part a) perform a Gram-Schmidt procedure on the ordered basis 0 0 S,,, 0 0 0, to obtain an orthonormal basis for M restricted to real entries.. (4 points) π a) Using ( uv, ) u( tvt ) ( ) dtas the inner product on C [ 0, ] S, for W span,cos () t,sin () t 0 { } π, construct and display an ordered orthonormal basis, 6/30/03 Madison Area Technical College

60 Page 60 b) Give the S coordinate representations of the following: [ ] S cos( t) S sin( t) S c) Determine the function in W closest to f () t t in C[ 0, π ]. d) What is the minimum angle between any function in W and f () t t in C[ 0, π ].. ( points) Decompose the matrix A into its QR factorization. 0 0 A Q R 6/30/03 Madison Area Technical College

61 Page 6 3. (3 points) Prove that if the Gram Schmidt process is applied to any set of n vectors in an inner product space then the zero vector is produced if and only if the set of n vectors is linearly dependent ( points) In R 4 determine an orthonormal basis for the orthogonal complement of W span,,. 6/30/03 Madison Area Technical College

62 Page 6 x 5. ( point) Let W y x y+ z 0. Determine the projection of z α β onto W. γ 6. (3 points) T a) Prove that the orthogonal complement of the null space of an m x n matrix A is the column space of A. b) What can you conclude about the orthogonal complement of the null space of T A? 6/30/03 Madison Area Technical College

63 Page (3 points) a) Using ( uv, ) ( ) ( ) u tvt dtas the inner product on P, find a non zero element of W, the orthogonal complement 0 of the span { t, t } +. Determine the following: b) ProjW () c) Proj W ( t ) d) Proj W () t e) Explain why the answers to b) and c) should be obvious. 6/30/03 Madison Area Technical College

64 Page (4 points) Define ( uv, ) u( tvt ) ( ) dtas the inner product on C[,]. Let W ( πt) ( πt) a) Produce and display an orthonormal basis for W. { } span cos, sin,. b) Determine g ( x ), the function which is the projection of ( ) f x x + x onto W. c) In Winplot generate and hand in two separate plots. The first plot graphs both f ( x) and ( ) plot graphs f ( x) g( x),. on[ ] g x on[,]. The second 9. ( points) Prove that if A is an m x n matrix with a column rank of n, then the solution of Ax b in the least T squares sense is give by x R Q bwhere R and Q are the matrices in the QR factorization of A. 6/30/03 Madison Area Technical College

65 Page ( points) Consider the following linear system of equations: x + y + 5z + w x + y z + w x 3y z 4w 9.73 x 4y + z w x + y + 3z + w x + y 3z + 8w 47.7 a) Does this system have any solutions? Explain your answer. b) Determine the solution of this system of equations in the least squares sense of solution.. (5 point Bonus Question) Consider the quadratic approximation to fitting a data set ( xi, yi) with i running from to n. β0 + βxi + βxi yi a) In general for n > 3 this system of equations has no solution for β0, β, β.set up the solution for the least squares solution of this problem. 6/30/03 Madison Area Technical College

66 Page 66 b) For the following set of 9 data points determine the values of β0, β, βthat minimize the sum n i ( ) yi β0 βxi βxi. i x i y i c) Generate and hand in a plot that shows both the original data as well as the quadratic fit. 6/30/03 Madison Area Technical College

67 Page 67 Project 5 Due 4//04 Score /48 Name. (4 points) Indicate which of the following transformations are linear transformations. Explain your reasoning in each case. a) a b a d L: M R L + c d b + c b) a 3 a L: R R L b b c) a 3 a L: R R L b b 0 L: M33 M3 L A A d) ( ) 3 π L C R L f x f t t dt π e) : ( ππ, ) ( ) () cos() f) L: R3 R L( [ a, b, c] ) ( + c) ab 5a 6/30/03 Madison Area Technical College

68 Page 68 g) L: R3 R3 L( [ a, b, c] ) [ a+, b+, c+ ] h) L: C (, ) C(, ) L f ( x) 3f ( x) + 7f ( x) i) L C ( ) C( ) L f ( x) f ( x) :,, x L C C L f x f t dt 0 j) : (, ) (, ) ( ) ( ). (4 points) The Linear Algebra Toolkit site Finding the kernel of the linear transformation allows you to define a linear transformation L from a vector spaces V to a vector space W for selected choices of V and W. The program will display a sequence of row operations that determines a basis and hence the dimension of the kernel of the linear transformation. The related Linear Algebra Toolkit site Finding the range of the linear transformation will display a sequence of row operations that determines a basis and hence the dimension of the range or image of the linear transformation. Use both of these programs to analyze each of the following linear transformations. Printout and attach the output for just parts f) and g). a) V R 4, W R 4 c c 3 5 c c L c 9 3 c c 4 c 4 A basis for the kernel of L A basis for the Image of L The nullity of L (the dimension of the kernel of L) Is L a one-to-one transformation? The rank of L (the dimension of the range of L) The nullity of L + the rank of L 6/30/03 Madison Area Technical College

69 Page 69 b) V R 4, W R 4 c c 7 5 c 4 3 c L c c c 4 c 4 A basis for the kernel of L A basis for the Image of L The nullity of L (the dimension of the kernel of L) Is L a one-to-one transformation? The rank of L (the dimension of the range of L) The nullity of L + the rank of L c) V R 4, W R 3 c c 3 6 c c L 5 3 c3 7 8c3 c 4 c 4 A basis for the kernel of L A basis for the Image of L The nullity of L (the dimension of the kernel of L) Is L a one-to-one transformation? The rank of L (the dimension of the range of L) The nullity of L + the rank of L d) V R 3, W R 4 5 c 7 3 c L c c 5 8 c c 3 8 A basis for the kernel of L A basis for the Image of L The nullity of L (the dimension of the kernel of L) Is L a one-to-one transformation? The rank of L (the dimension of the range of L) The nullity of L + the rank of L 6/30/03 Madison Area Technical College

70 Page 70 e) V M, W M c c c c3 c c4 L + c 3 c 4 c c 4 c c + 3 A basis for the kernel of L A basis for the Image of L The nullity of L (the dimension of the kernel of L) Is L a one-to-one transformation? The rank of L (the dimension of the range of L) The nullity of L + the rank of L f) V P 4, W P 3 A basis for the kernel of L L is the first derivative operator A basis for the Image of L The nullity of L (the dimension of the kernel of L) Is L a one-to-one transformation? The rank of L (the dimension of the range of L) The nullity of L + the rank of L g) V P 4, W P 5 t p x dx for p(t) any polynomial in P 4 0 [ ()] ( ) L p t A basis for the kernel of L A basis for the Image of L The nullity of L (the dimension of the kernel of L) Is L a one-to-one transformation? The rank of L (the dimension of the range of L) The nullity of L + the rank of L 6/30/03 Madison Area Technical College

71 Page 7 3. ( points) For each linear transformation below determine the nullity (the dimension of the kernel) and the dimension of the image (range). L: R R L abc,, ab, c a) 3 ([ ]) [ ] L M M L A A b) ( ) : c) L: P P L f ( x) f ( x) 3 d dx 5 3 L R R L x x d) : ( ) 4. ( points) Given a fixed vector u in n inner product define the function fu ( v) ( u, v) a) Compute f ( u ) u R defined component wise by [ ] for all v R. n ( ) j u j,, n, using the standard j n b) Does f ( ) u v define a linear transformation? If so from what vector space to what vector space? c) Consider the surface of a sphere of radius 3 in n of n R? n R, i.e., A { v v R and v 3}. Explain whether A is a subspace d) What is the maximum value of ( ) u f v on A? For what v Ais this maximum attained? 6/30/03 Madison Area Technical College

72 Page 7 5. (3 points) A is an m x n matrix of rank n. a) What is the minimum value of m the number of rows of A? b) What is the dimension of the ker( A ) (the null space of A)? T c) What is the dimension of the ker( ) A A? d) Is T A Asingular or non singular? Prove your answer. e) Prove that T A A is a positive definite matrix. k 6. ( points) (, ) derivatives. Note: C (, ) illustrates. f ( x) C is the vector space of real valued functions defined on(, ) with continuous k th order 0 if x 0 x sin if x 0 x continuous at the origin. a) Explain whether the first derivative D transformation on C (, ) is only a proper subset of the set of differentiable functions as the following example has the derivative f ( x) d dx, what is the range? 0 if x 0 which is not xsin cos if x 0 x x is a linear transformation on (, ) C. If Dis a linear b) Explain whether the k th derivative k transformation on C (, ) Dk k d k dx, what is the range? k is a linear transformation on (, ) C. If Dk is a linear 6/30/03 Madison Area Technical College

73 Page 73 and let L f ( x) D f ( x) 5D f ( x) + 6f ( x) Let Wλ span{ e λt } c) Explain whether W λ is a subspace of (, ). C. d) Prove that L is a linear operator on W λ. e) Display the representation of L in the basis S { e λt }. 7. (3 points) Prove that the following three statements are equivalent. a) The linear transformation LV : Wis one to one. b) The dimension of the kernel of L is zero. c) For every basis S { v v v } of V, LS ( ) Lv ( ) Lv ( ) Lv ( ),,, n {,,, n } is a basis for the image of L. 8. ( points) Given the linear transformation LV : W if L is onto. Prove that if dim(v) dim(w), then L is one to one if and only 6/30/03 Madison Area Technical College

74 Page (3 points) a) Prove that the linear transformation LV : W is invertible if and only if L is an isomorphism. b) Prove that the inverse of an isomorphism is an isomorphism. c) LV : W is an isomorphism, S { V, V,, Vn} is a basis for V, T { w w w },,, n is a basis for W, and A is the n x n matrix that represents L with respect to the S, T basis. Prove that the matrix that represents respect to the T, S basis is A. L : W V with 0. (6 points) Given the linear transformation L: M 3 R defined by a+ d a b L b c + c d and the ordered bases a d S,,,,,, S T 0,, 0 T 0,, /30/03 Madison Area Technical College

75 Page 75 a) Determine the matrix that represents the identity operator on M with respect to S and S. b) Determine the matrix that represents the identity operator on 3 R with respect to T and T. c) What is the dimension of the kernel of L? d) What is the dimension of the image or range of L? e) Determine the matrix that represents L with respect to S and T. f) Determine the matrix that represents L with respect to S and T.. (9 points) df Given the linear transformation D : P3 Pdefined by D( f () t ) and the ordered bases dt 3,,, 3, 3,,,,, S t t t S t + tt tt + t T t t T t + t, t { } { } { } { } a) Determine the matrix that represents the identity operator on P 3 with respect to S and S. 6/30/03 Madison Area Technical College

76 Page 76 b) Determine the matrix that represents the identity operator on P with respect to T and T. c) What is the dimension of the kernel of D? d) What is the dimension of the image or range of D? e) Determine the matrix that represents D with respect to S and T. f) Determine the matrix that represents D with respect to S and T. d d Now consider the linear operator L: P3 P3 dt d g) Determine the matrix that represents L with respect to S and S. defined by L f () t f () t 5 f () t + 6f () t h) Determine the matrix that represents L with respect to S and S. i) Determine the matrix that represents L with respect to S and S. j) Determine the matrix that represents L with respect to S and S. 6/30/03 Madison Area Technical College

77 Page 77. (3 points) V and W are vector spaces and U VW is the vector space of all linear transformations from V to W. In a) through d) below state the dimension of U VW. 3 a) V R W R 7 b) 7 V R W R 3 c) V P W P d) V M W R e) Consider the following three linear transformations from R R u u u u L u+ u u u, L 0 u u and L 3 u u u. In the space U of all linear transformations of u u L, L, L linearly independent? the set { } 3 3 R R is 3. ( points) L is a linear operator LV : product designated as (, ) from a vector space V to itself, and for all u and v in V there is an inner Lu, Lv uv, Lu u. V ( ) ( ) uv. Prove that ( ) ( ) for all u and v in V if and only if ( ) 6/30/03 Madison Area Technical College

78 Page (3 points) a) Prove that on Mnn similarity is an equivalence relation, i.e., similarity is reflexive, symmetric and transitive. b) Prove that similar matrices have the same rank. c) Prove that if matrix A is similar to matrix B, then for any whole number k, k A is similar to B. k 5. (4 point Bonus) 0 0 A B A and B represent linear transformations on the homogeneous coordinates x y. a) Describe the result of the transformation represented by A b) Describe the result of the transformation represented by B. c) Describe the result of the transformation represented by AB. d) Describe the result of the transformation represented by BA. 6/30/03 Madison Area Technical College

79 Page 79 e) Sketch the result of the transformation represented by AB on the triangle shown in the figure below. f) Sketch the result of the transformation represented by BA on the triangle shown in the figure below. 6/30/03 Madison Area Technical College

80 Page 80 Group Project 4 Due 4/30/04 Score / Name Name Name Name Choose one of the following three problems. Each problem is worth points. You may do extra problems to earn up to 4 extra credit points.. Fourier Series: Part I. Fejer s theorem essentially asserts that the set of functions { } basis for continuous functions on[ 0,a ]. πx j j a j πx sin j cos a constitute a a f g f x g x dx 0 πx j j a j a) Define the inner product of two continuous functions on [ 0,a ] as (, ) ( ) ( ) construct and display an orthonormal basis from { } πx sin j cos a. Using this inner product 6/30/03 Madison Area Technical College

81 Page 8 b) On [0, a] let f ( x) bx a if 0 x a x a b if < x a a Generate and display the infinite Fourier series that represents f(x) in terms of the orthonormal basis of part a). c) In Winplot define f(x) with User functions from the Equa menu. The following functions built into Winplot will aid you in this section. m sum ( f ( x, j), j, k, m ) f ( x, j) j k floor(x) is the greatest integer function defined as the greatest integer less than or equal to x. For example, floor(.789), floor(3) 3 and floor(-.005) -3. if x 0 hvs(x) is the Heaviside function defined as hvs( x). This can be used as a switch to generate a piece 0 if x < 0 wise function. This is illustrated below. k( x) if x a Let g( x). Then g( x) h( x) + k( x) h( x) hvs( x a) h( x) if x< a. x a extends f(x) to the entire real number line and generates a triangle wave of period a which matches the period of the trigonometric basis. The function F( x) f x afloor For a 3 and b plot F(x) and the first three non-zero terms of the Fourier series on the same graph from at least -.5a to.5a. Print and attach your plot. Prepare, print out and attach a second plot that displays the error, f(x) Fourier sum, on [0, a]. Such a graph is illustrated below. Finally, repeat this whole process (i.e., generate and hand in two more plots) for the first eleven non-zero terms of the Fourier series. 6/30/03 Madison Area Technical College

82 Page 8. Fourier Series: Part II. Consider the projection of the function f(x) into the subspace spanned by the four functions πx 4πx 6πx, cos, cos, cos a a a. This projection can be expressed as follows: πx 4πx 6πx G( x) β0 + βcos βcos β3cos a + + a a The coefficients are chosen to minimize the squared distance from f(x) to G(x). Specifically, a D( β0, β, β, β 3) ( f ( x) G( x), f ( x) G( x) ) where the inner product ( f, g) f ( x) g( x) dx. 0 a) Determine the values of β0, β, β, β 3 that minimize D( β0, β, β, β 3). 6/30/03 Madison Area Technical College

83 Page 83 b) A second way to approach this least squares approximation which would be especially appropriate if f(x) were defined by data rather than a formula is to consider the following over determined system of equations. β0 f ( x ) β f ( x ) A β β 3 f ( xn ) For a suitable choice of n sampling points x, x, xn the n x 4 matrix A defined as π 4π 6π cos x cos x cos x a a a π 4π 6π cos x cos x cos x A a a a. π 4π 6π cos xn cos xn cos xn a a a. f ( x ) f ( x ) This matrix has rank 4 and in general for n > 4 the range of A will not contain so that the system of f ( xn ) equations has no solution. However, there is a least squares solution that minimizes the distance from β0 f ( x ) β0 f ( x ) β A f ( x ) to β. It is given by T T f ( x ) β ( A A) A. β β 3 f ( xn ) β 3 f ( xn ) ja For a 3 and b let n 50 and let x j with 0 j n. For these values use a spread sheet to construct the n T matrix A A and then solve for β0, β, β, β3. If the simplicity of T A A surprises you it can be explained by Euler s iθ + cos( s) identity e cos( θ) + isin ( θ), the trigonometric identity cos ( s) and the geometric n n j r series r! Hand in your spread sheet and include in it two plots. One that shows both f(x) and the r j 0 minimizing G(x) on [0, a] and the second which shows the error (the residuals ), f(x) - G(x), on [0, a]. c) How do the estimates β0, β, β, β3 compare to β0, β, β, β 3 from part a). Explain any discrepancy. 6/30/03 Madison Area Technical College

84 Page Legendre Polynomials On the interval [-, ] the polynomials { } { x } j j form a basis for all continuous functions on [-, ]. One can construct a sequence of orthonormal polynomials with respect to the inner product (, ) ( ) ( ) polynomials are called the normalized Legendre Polynomials. Let L ( ) Polynomial of degree n for which the coefficient on x n is positive n f g f x g x dx. These x designate the normalized Legendre a) Produce the first five normalized Legendre Polynomials. Wolfram Alpha can help with the integrals! n L ( x ) n b) Show that each of the normalized Legendre Polynomials you produced solves the following differential equation. ( ) ( ) x y xy + n n+ y 0 6/30/03 Madison Area Technical College

85 Page 85 c) Does Ln ( x) contain any nonzero even powered terms for odd n? Does Ln ( ) terms for even n? Prove your answers. x contain any nonzero odd powered d) On the interval [-a, a] the polynomials { } { x } j j form a basis for all continuous functions on [-a, a]. In analogy with the Legendre Polynomials one can construct a sequence of orthonormal polynomials on [-a, a] with respect to the inner product ( f, g) f ( x) g( x) dx. Let ( x ) a a n designate such a normalized polynomial of degree n on [-a, a] for which the coefficient on x n is positive. Produce the first five such normalized polynomials. n ( x ) n e) The function e x on the interval [-, ] can be expanded as an infinite series of normalized Legendre Polynomials. x Specifically, e α L ( x) j j. Determine j j 0 n α for 0 j 4. Again Wolfram Alpha can help with the integrals. α n 6/30/03 Madison Area Technical College

86 Page 86 x f) In Winplot generate, print and attach a plot which shows the error, e α L ( x) 4, on the interval [-, ]. Defining each of the five Legendre Polynomials in User functions from the Equa menu will make this easier. j 0 j j Bonus: (3 points each part) Attach work on separate sheets. g) Define the n th degree polynomial, ( ) n considering even and odd n separately that ( ) n n d p x, by pn ( x) ( x ) n. Show by using the binomial theorem and n dx x y xy + n n+ y 0. p x is a solution of ( ) ( ) h) Show by induction that for j n d j n n j, ( x ) ( x ) g ( x) dx j where g ( ) nj, nj, x is a polynomial. i) Use the result of part h) to show that ( f, pn) f ( x) pn( x) dx 0for f ( ) n. x any polynomial of degree less than j) Use the result of part i) to prove that p ( x) c L ( x) n n n for some non zero constant n c. k) Determine an explicit formula for c n as a function of n. 6/30/03 Madison Area Technical College

87 Page 87 Project 6 Due 5/08/04 Score /48 Name. (3 points) Go to AKiTi Miscellaneous Mathematical Utilities page at and numerically determine the eigenvalues and eigenvectors of the following matrices. Fill in each table below. a) 3 A det[ A ] λ λ λ 3 λλλ 3 b) A det[ A ] λ λ λ 3 λ 4 λλλλ 3 4. (3 points) Let L be a linear operator from a finite dimensional vector space V to itself. The eigenspace of L associated with a given complex numberλ is defined as Wλ w w V and L( w) λw. a) Is the zero vector in W λ? { } b) Show that W λ is a subspace of V. c) Prove that L is an isomorphism if and only if W 0 { 0 V } d) A is 3 3 real symmetric matrix with eigenvalues λ, λ, λ 3 and associated eigenvectors X, X, X 3. If λ λ and v is a nonzero vector in the orthogonal complement to the span of X 3, compute Av v. 6/30/03 Madison Area Technical College

88 Page (3 points) A is an n x n real matrix with n eigenvaluesλλλ,, 3,, λ n and associated eigenvectors x i, Axi λix i, x 0. i a) If no two of the eigenvalues are the same must the eigenvectors be linearly independent? Prove your answer. b) If no two of the eigenvalues are the same must A be nonsingular? Support your answer with an argument. c) Must all of the eigenvalues be real? Support your answer with an argument. 4. ( points) a) What can you conclude about a x real symmetric matrix that has only one distinct eigenvalue? b) Prove that a x real matrix is defective if and only if it is non-symmetric and has only one distinct eigenvalue. 6/30/03 Madison Area Technical College

89 Page ( points) a) Prove that the transpose of a matrix has the same eigenvalues as the original matrix. Is the same true of the eigenvectors? Explain your answer. b) For real symmetric matrices the eigenvectors can always be expressed using only real numbers. Is this true of any real square matrix? Explain your answer. 6. ( points) a) Prove that the determinant of any square matrix is the product of its eigenvalues. b) Prove that the trace of any non defective matrix is the sum of its eigenvalues. [This result is actually true for any square matrix!] 7. ( points) Determine the eigenvalues and associated eigenvectors of the following matrices: a) 3 A 8 b) 5 A 5 5 6/30/03 Madison Area Technical College

90 Page 90 c) 4 A 3 d) + i A 0 Hint: ( ) ( ) i cos cos e θ + θ + i θ e) 0 4 A /30/03 Madison Area Technical College

91 Page 9 f) A (3 points) Indicate which of the following matrices can be diagonalized (are not defective). Justify your answers. a) A b) B c) C /30/03 Madison Area Technical College

92 Page 9 9. (3points) A is a real symmetric matrix with a) Evaluate A 8 A ; A 0 0 ; A b) Evaluate A c) Evaluate a A b c 0. (7 points) Complex Inner Product Spaces, Unitary and Hermitian matrices. The concept of an inner product can be extended to vectors consisting of complex entries. In analogy with R n, the n space of n dimensional vectors with complex components is called. The inner or Hermitian product of two n n T vectors in is defined as u v ( u, v) u v ujvj. Here z designates the complex conjugate of z. Recall 6/30/03 Madison Area Technical College j that for a complex number z, z a+ biwith a and b real, the complex conjugate is defined as z a bi. z zz a+ bi a bi a + b 0. The convention of complex conjugation on the right in the definition of u v is arbitrary and some authors define the Hermitian product with complex conjugation on the 5+ i left. What s important is to be consistent. As an example of the Hermitian product, let u 3 and 4i 3+ 7i v i, then u v ( 5+ i)( 3 7i) + ( 3)( i) + 4i( + i) 5 3i. i Furthermore, ( )( )

93 Page 93 a) Show that the Hermitian inner product satisfies the following properties. Here u, v, and w represent vectors and c is a complex scalar. u v v u u+ v w u w + v w w u+ v w u + w v cu v c u v u cv c u v u u 0 with equality if and only if u 0. The Cauchy Schwartz inequality is still true. For any complex scalarα and non-zero vectors u and v, u+ αv u+ αv 0. So, u+ αv u+ αv u u + α v u + α v u + α v v 0. Let α v u v u v u u u v v u u 0 v v + v v v v. Rearranging this inequality gives u u v v v u with equality if and only if u cvfor some complex scalar c. u v v v, then T The adjoint or Hermitian conjugate of an n x n complex matrix is defined as A A, i.e., the transpose of the matrix of complex conjugates. Note: when moving a matrix multiplier from left to right in an inner product one transforms to Au v Au v u A v u A v u A v T T T T T the Hermitian conjugate. ( ) ( ). An n x n complex matrix U is called unitary if and only if U matrix A is called Hermitian if and only if A A. U, i.e., UU UU I n. An n x n complex 6/30/03 Madison Area Technical College

94 Page 94 b) Which of the following matrices are unitary? 3+ i i 0 0 A + i + i 0 0 i i i i 0 0 B i + i 0 0 i i i 0 0 i 0 0 C + 3i i i i 0 0 ( ϕ ) ( θ) ( θ) i ( ϕ ) ( θ ) ( ϕ ) ( θ) ( θ) i ( ϕ ) ( θ ) i cos( ϕ) 0 sin ( ϕ) sin cos sin cos cos D sin sin cos cos sin c) What is true about the determinant of a unitary matrix? d) Is every orthogonal matrix a unitary matrix? Explain your answer. 6/30/03 Madison Area Technical College

95 Page 95 e) Look up the definition of a group in algebra. Show that the set of n x n unitary matrices form a group with respect to the operation of matrix multiplication. f) Let SU(n) be the set of all n x n unitary matrices with a determinant of one. Show that SU(n) is a group with respect to the operation of matrix multiplication. Find an element of SU() that is not the identity matrix. g) Suppose A is an n x n matrix and U is an n x n unitary matrix. What is the det U AU? What is the trace U AU? h) Can a non-identity matrix be both Hermitian and unitary? Explain your answer. 6/30/03 Madison Area Technical College

96 Page 96 i) For any matrix below which is Hermitian determine its eigenvalues i 5 3i 5 3 5i i 5i 3 5i 5i j) Suppose A is n x n Hermitian matrix and u is a vector in n. What is Au u Au u? k) Suppose A is n x n Hermitian matrix and for a non-zero ϕ, Aϕ λϕ. What is λ λ? Choose two of the following five problems. Each problem is worth 4 points. You may do extra problems to earn up to extra credit points.. Let A T a) Determine the eigenvalues and an orthonormal set of eigenvectors of the matrix AA. 6/30/03 Madison Area Technical College

97 Page 97 b) Determine the eigenvalues and an orthonormal set of eigenvectors of the matrix T AA. c) Determine the singular value decomposition of A. d) State and prove a conjecture about the relationship of the set of eigenvalues of T AA. T AA to the set of eigenvalues of 6/30/03 Madison Area Technical College

98 Page 98. Iterative Scheme to Calculate Eigenvalue of Greatest Absolute Value Consider a real n x n matrix A with n distinct real eigenvalues. Let E be the most extreme eigenvalue (i.e., the eigenvalue of largest absolute value) of A with eigenvector Φ and designate the remaining eigenvectors of A n as{ Φ k } k with AΦ k λkφ k. Let Ψ be a vector in n R. Because the eigenvectors of A form a basis for n R, Ψ has an expansion in the eigenspace of A: Ψ cφ + ckφk. Assume that ck 0. n n n n n m m m m Then ( A ΨΨ, ) A ckφk, cjφ j E c cj ( Φ, Φ j) + λk ckcj ( Φk, Φ j). k Since E n k k j j k j + ( A, ) m ( A ΨΨ, ) n n n m ( Φ, Φ ) + λ ( Φ, Φ ) E c c c c m+ + m j j k k j k j j k j ΨΨ n n n m j ( Φ, Φ j) + λk k j ( Φk, Φ j) E c c c c m j k j m+ m+ λ j k E j m m λ E c c c c j k E j n n n k j ( Φ, Φ j) + k j ( Φk, Φ j) E c c c c n n n k j ( Φ, Φ j) + k j ( Φk, Φ j) m+ λ c c c c j k E j E m λ c c c c j k E j m n n n k j ( Φ, Φ j) + k j ( Φk, Φ j) n n n k j ( Φ, Φ j) + k j ( Φk, Φ j) λ <, all of the sums n n λk c (, ) k cj Φk Φ and n n λk ( ) j c k cj Φk, Φ will approach zero j E k j ( m A + ΨΨ, ) as m increases. Thus as m increases the ratio m ( A ΨΨ, ) eigenvalues and fill in the following tables. E m+ k j converges to E. For each matrix given determine all of its 6/30/03 Madison Area Technical College

99 Page 99 a) 7 8 A E Ψ λ λ m m ( A + ΨΨ, ) m ( A ΨΨ, ) b) A 6 4 E Ψ λ λ m m ( A + ΨΨ, ) m ( A ΨΨ, ) /30/03 Madison Area Technical College

100 Page 00 c) A 6 4 E m m ( A + ΨΨ, ) m ( A ΨΨ, ) Ψ d) 5 0 A E m m ( A + ΨΨ, ) m ( A ΨΨ, ) Ψ /30/03 Madison Area Technical College

101 Page 0 e) A E m m ( A + ΨΨ, ) m ( A ΨΨ, ) Ψ f) A E m m ( A + ΨΨ, ) m ( A ΨΨ, ) Ψ /30/03 Madison Area Technical College

102 Page 0 g) A E m m ( A + ΨΨ, ) m ( A ΨΨ, ) 3 4 Ψ h) A E m m ( A + ΨΨ, ) m ( A ΨΨ, ) 4 3 Ψ /30/03 Madison Area Technical College

103 Page 03 i) Why are the results for b) and c) different? j) Explain the results seen in e) through h. 3. For each of the following systems of differential equations: i) Obtain the general solution that solves the initial condition ( 0) ( 0) x x0 y y. 0 ii) Determine and classify all equilibrium points in R. iii) In Winplot generate and attach a labeled plot that displays the phase portrait of the system, marks the equilibrium points, and graphs the particular solution with x0 3 y 0. To generate the plot access Winplot -dim, from the Equa menu select Differential. dx/dt. Enter the particular solution as Parametric equations from the Equa menu. Additional solutions can be displayed by using the menu sequence One/Initial-value problems. 6/30/03 Madison Area Technical College

104 Page 04 a) dx dt dy dt 3x + 9y x 5y b) dx dt dy dt 4x + 4y 5x 4y c) dx dt dy dt 0x 9y x 4y d) dx dt dy dt x + y x 4y 6/30/03 Madison Area Technical College

105 Page Suppose A is a real n x n matrix and x n R. a) Can the linear system dx Ax have more than one equilibrium point? Explain your answer. dt b) Can the linear system dx Ax have exactly two equilibrium points? Explain your answer. dt c) Suppose A is a real x defective matrix, explain how you can obtain the general solution of d x x A dt y y that solves the initial condition ( 0) ( 0) x x0 y y 0. Hint: Consider the x matrix U [ U, U ], where U is a normalized eigenvector of A and U is a normalized vector orthogonal to U. Now construct the orthogonal transformation T U AU and define C T x U C y. 6/30/03 Madison Area Technical College

106 Page 06 d) Apply the method of part c) to solve dx 5x + y dt x( 0) x0 with dy y ( 0 ) y. 0 4x + y dt 5. The Fibonacci Sequence a) Consider the rectangle ABCD with AE AD and rectangle FEBC similar to rectangle ABCD. Determine the exact value of AB φ. AD 6/30/03 Madison Area Technical College

107 Page 07 b) The Fibonacci sequence,,,3,5,8,3,,34,55 is defined recursively as F F and Fn+ Fn+ + Fnfor n. This can be expressed in matrix form as X n+ AX, where Fn + n X n F and n A 0. Note that A is a symmetric matrix. Determine the eigenvalues λ and λ and an associated pair of orthonormal eigenvectors, V ˆ and V ˆ of the matrix A. c) Let X C ( n) Vˆ + C ( n) Vˆ. Determine explicit formulas for () n C and C () in terms of φ. 6/30/03 Madison Area Technical College

108 Page 08 d) Determine the recurrence relation for C ( n+ ) in terms ofφ and C ( ) C ( n+ ) in terms ofφ and ( ) C n. n and the recurrence relation for e) Obtain explicit formulas for C ( n) and ( ) C n. f) Express Fn as in explicit function of n in terms of φ. 6/30/03 Madison Area Technical College

109 Page 09 Group Project 5 Due at Final Exam Score / Name Name Name Name Choose one of the following 5 problems. Each problem is worth points. You may do extra problems to earn up to 48 bonus points.. The Spectral or Fourier Representation of Hermitian Matrices a) If A is an n x n Hermitian matrix then it has n linearly independent eigenvectors which span n. Let n Φ j designate a set of n normalized linearly independent eigenvectors of A with AΦ j λ jφ jand Φ j Φ j. { } j If all of the eigenvalues of A are distinct, determine Φ j Φ k. Can this result still be true even if the eigenvalues of A are not all distinct? Explain your answer. b) Let B be an n x n matrix defined as Bij i'th component of eigenvalues of A are distinct determine BB. Φ, i.e., B [ Φ Φ Φ ] j n. If all of the Wolfram Alpha can be used to determine the eigenvalues and a set of eigenvectors for a matrix as illustrated below. 6/30/03 Madison Area Technical College

110 Page 0 c) Determine the eigenvalues and a set of orthonormal eigenvectors of the matrix A. 5 3 i A 3+ i λ Φ λ Φ Let v i. Compute the following: j λ v Φ Φ Av j j j d) Determine the eigenvalues and a set of orthonormal eigenvectors of the matrix A. 3 3i i A i 3i λ Φ λ Φ λ Φ 3 3 6/30/03 Madison Area Technical College

111 Page Let i v i. Compute the following: 3 j λ v Φ Φ Av j j j e) Let A be an n x n Hermitian matrix with an orthonormal set of eigenvectors { j} j Φ Φ δ. Let v be any vector in j k jk n n. What does the sum λ j v Φ j Φ j represent? j n Φ so that AΦ j λ jφ jand n If A is nonsingular, what does the sum v Φ j Φ j represent? λ j j Problems and 3 are dedicated to the memory of Professor C. H. Blanchard (93 009) of the University of Wisconsin-Madison Physics Department. He was the best and most inspiring teacher I have ever encountered.. The Moment of Inertia Tensor In the analysis of the rotation of a rigid body one defines a second rank tensor, called the moment of inertia tensor which can be represented by the following 3 x 3 real symmetric matrix. For a mass density ρ ( x, yz, ) m I I I xx xy xz inertia xy yy yz I I I I I I I xz yz zz triple integrals over the region of three dimensional space S bounded by the surface of the rigid body. 6/30/03 Madison Area Technical College, the matrix elements of the moment of inertia tensor are defined by the following

112 Page xx xy xz yy yz zz S S S m S S m S m (,, )( ) I ρ xyz y + z dv I I m m (,, ) (,, ) (,, )( ) m x y z xydv x y z xzdv I ρ x y z x + z dv I ρ ρ ρ (,, ) x y z yzdv (,, )( ) I ρ xyz x + y dv ωx For an angular velocity vector ω ω y, the angular momentum of the rigid body is given by ω z Iinertiaω and the kinetic energy of rotation is ω T Iinertiaω. For a uniform solid right circular cylinder of radius a and height h inclined as shown in Figure, the moment of inertia tensor is h 5a 3 h a h a Iinertia h a h 7a a) Determine the eigenvalues and an orthonormal set of eigenvectors for this I inertia. 6/30/03 Madison Area Technical College

113 Page 3 Figure b) The orthonormal eigenvectors are called the principal axes of the rigid body. Explain this terminology. 6/30/03 Madison Area Technical College

114 Page 4 3. Coupled Pendulums A simple pendulum consists of an object of mass m suspended by an essentially weightless and non stretchable string of length that is allowed to swing freely. As in figure let the angleψ denote the displacement from the downwards 3π vertical. ψ is related to the standard polar angleθ byθ + ψ, so that the position vector of the mass is given by 3π ˆ 3π cos sin ˆ R ψ i ψ j sin ( ψ) iˆ cos( ψ) ˆ j. The acceleration vector is thus d R ( ) ˆ ( ) ˆ dψ sin cos cos( ) ˆ sin( ) ˆ d ψ a ψ i ψ j ψ i ψ j dt dt dt dψ d ψ ( ) ( ) ˆ dψ d ψ sin ψ cos ψ i cos( ψ) sin ( ψ) ˆ j dt dt dt dt Figure 6/30/03 Madison Area Technical College

115 Page 5 The forces acting on the mass are the tension in the string, T, which points to the origin and gravity which points straight down. F T sin( ψ) iˆ+ cos( ψ) ˆj mg ˆj Tsin ( ψ) iˆ+ Tcos( ψ) mg ˆ j. From Newton s second law of F ma m d ψ d ψ i m d ψ d ψ ˆ j dt dt dt dt motion, sin ( ψ) + cos( ψ) ˆ+ cos( ψ) + sin ( ψ) coupled differential equations, so that we have the dψ d ψ Tsin( ψ) m sin( ψ) + cos( ψ) dt dt dψ d ψ Tcos( ψ) mg m cos( ψ) + sin ( ψ) dt dt ( ) dψ cos ψ d ψ From the first equation T m dt. Substituting this expression into the second equation gives sin ( ψ ) dt the following. ( ψ ) ( ψ ) dψ cos d ψ dψ d ψ m cos( ) mg m cos( ) sin ( ) dt ψ ψ sin dt + ψ dt dt cos ( ψ ) d ψ d ψ g sin ( ψ ) sin ( ψ ) dt dt d ψ d ψ sin ( ψ) cos ( ψ) g sin ( ψ) dt dt d ψ g sin( ψ ) dt We seek solutions that satisfy the initial values of ψ ( 0) and ψ ( 0). This non linear second order initial value problem has a solution which can be expressed as an elliptic function. A simpler result is obtained if we restrict our attention to small angles where ( ) sin ψ ψ. The differential equation then is linear and is given by d ψ g ψ. dt 6/30/03 Madison Area Technical College

116 Page 6 t Looking for exponential solutions of the form ψ () t e λ g gives the auxiliary equation λ which has solutions g λ ± i ± iω0 iω0t iω0t. Thus, ψ () t be b e ( b b ) cos( ω t) i( b b ) sin ( ω t) solution be real means that ψ () t ψ () t Requiring that the 0 0, so that ( b b ) cos( ω t) i( b b ) sin ( ω t) ( b b ) cos( ω t) i( b b ) sin( ω t) Since sine and cosine are linearly independent functions it must be true that b+ b b+ b b b b b. Taking c ic, d + b b id for real c, c, d, d yields d cand d cso that b + b c and b b ic. So the general solution in the small angle approximation is ψ () t c cos( ω t) c sin ( ω t) the initial conditions gives () ( 0cos ) ( ) ( 0) ( ) 0 0 ω0 +. Fitting 0 0 ψ ψ t ψ ω t + sinω t. The solution is periodic with a period given by π π g T and a frequency of f ω g T π 0 () t ( 0cos ) ( t), so that ψ ( 0) ψ ψ ω 0. If the pendulum is released from rest, ψ ( 0 ) 0and is the amplitude of the angle oscillations. Two pendulums each of length are coupled by a cross string as shown on Figure 3. The effect of the cross string can be modeled by a force on pendulum by pendulum equal to k ( ψ ψ ) where k is a positive proportionality constant. By Newton s third law of motion the force on pendulum by pendulum is k ( ψ ψ ) angle approximation the two angles satisfy the following linear differential equation. g k k d ψ ψ dt ψ g ψ k k. In the small 6/30/03 Madison Area Technical College

117 Page 7 Figure 3 Graphic from page 7 of Nine Experiments for a Third Grade Hour, 009 Memorial Edition by C. H. Blanchard. Figure 4 6/30/03 Madison Area Technical College