Extra Problems for Math 2050 Linear Algebra I


 Sibyl Patterson
 1 years ago
 Views:
Transcription
1 Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as a scalar multiple of u = 8, as a scalar multiple of v =, and as a scalar multiple of w = Justify your answers Given u =, u = parallel vectors, u = 6 6, u 4 =, u 5 = 5 Find the indicated vectors [ [ (a) 4 + (b) Suppose u = [ [ a, v = 5 6 b and w = [ Find a and b if u v = w 5, list all pairs of 7 Suppose x, y, u, and v are vectors Express u and v in terms of x and y given x y = u and x + y = v [ [ 4 8 Given u = and v =, find u v and illustrate with a picture like that in Figure 6 9 Shown at the right are two nonparallel vectors u and v and four other vectors w,w,w,w 4 Reproduce u, v, and w in a picture by themselves and exhibit w as the diagonal of a parallelogram with sides parallel to u and v Guess (approximate) values of a and b so that w = au + bv w w u v w 4 w Express [ 7 7 Is it possible to express as a linear combination of [ [ and [ as a linear combination of [ and [? Explain c MMVIII Edgar G Goodaire, Dept of Mathematics and Statistics, Memorial University of Newfoundland
2 Extra Problems 4 Let v =, v = 5, and v = 6 Is a linear combination of v, v, v? Is it possible to express the zero vector as a linear combination of more than one way? Justify your answer [ 4 and [ 8 in 4 Suppose u and v are vectors Show that any linear combination of u and v is a linear combination of u and v 5 Let O(,), A(,), B(4,4), C(, ), D(4,), E(6,), F(, 4), G(6,), H(8,) be the indicated points in the xyplane Let u = OA be the vector pictured by the arrow from O to A and v = OC the vector pictured by the arrow from O to C Find u and v Then express the vector [ OD from to D as a combination of u and v For example, 6 OE = = u + v A 6 Suppose ABC is a triangle Let D be the midpoint of AB and E be the midpoint of AC Use vectors to show that DE is parallel to BC and half its length B D E C 7 Let A = (, ), B = (4, ) and C = (, ) be three points in the Euclidean plane Find a fourth point D such that the A, B, C and D are the vertices of a parallelogram and justify your answer Is the answer unique? 8 Prove that vector addition is closed and commutative and every vector u = negative u which has the property that u + ( u) = ( u) + u = [ x y has a 9 Show that any linear combination of and is also a linear combination of 6 and Explain how one could use vectors to show that three points X, Y, and Z are collinear (Points are collinear if they lie on a line) If u and v are parallel, show that some nontrivial linear combination of u and v is the zero vector (Nontrivial means that u + v = doesn t count!)
3 Extra Problems Let u = and v = Find a unit vector in the direction of u and a vector of 4 length in the direction opposite to v Susie is asked to find the length of the vector u = 4 7 She proceeds like this u = 7, so u = ( 8 7 ) 4 + ( 7 ) + ( 7 ) = Do you like this approach? 4 Find the angle between each of the following pairs of vectors Give your answers in radians and in degrees If it is necessary to approximate, give radians to two decimal places of accuracy and degrees to the nearest degree (a) u = [ 4, v = [ 4 (b) u = [ 4, v = [ 5 Find u v given u =, v =, and the angle between u and v is 45 6 Can u v = 7 if u = and v =? Justify your answer 7 Let u = 4, v =, and w = (a) Find u, u + v and w w (b) Find a unit vector in the same direction as u (c) Find a vector of norm 4 in the direction opposite to v 8 Let u and v be vectors of lengths and 5 respectively and suppose that u v = 8 Find (u v) (u v) and u + v [ [ 9 Let u = and v = Find all numbers k such that u + kv has norm If a nonzero vector u is orthogonal to another vector v and k is a scalar, show that u + kv is not orthogonal to u Given the points A(,) and B(,) in the plane, find all points C such that A, B, and C are the vertices of a right angle triangle with right angle at A, and AC of length Find a point D such that (,), (, ), (4, ) and D are the vertices of a square and justify your answer
4 4 Extra Problems Let u = 4, w = 5 and v = 7 4 Show that u is orthogonal to v and to w 4 Assuming all vectors dimensional, prove that the dot product is commutative and that v v = if and only if v = [ [ 5 Verify the Cauchy Schwarz inequality for u = and v = 5 6 Use the Cauchy Schwarz inequality to prove that (acos θ + bsin θ) a + b for all a,b R 7 Let a,b,c,d be real numbers Use the Cauchy Schwarz inequality to prove that (ac+ bd) (a + b )(c + d ) 8 Suppose vectors u and v are represented by arrows OA and OB in the plane, as shown Find a formula for a vector whose arrow bisects angle AOB 9 Let P be a parallelogram with sides representing the vectors u and v as shown Express the diagonals AC and u O B v B C u A D v DB of P in terms of u and v 4 Express the plane with equation x y + z = as the set of all linear combinations of two nonparallel vectors (There are many possibilities!) 4 Find the equation of each of the plane parallel to the plane with equation 4x+y z = 7 and passing through the point (,,) 4 Verify that u v = (v u) given u = and v = 4 4 Given u = and v =, find the cross product u v, then verify that this vector is perpendicular to u and to v and that v u = (u v) 4 44 Given u =, v =, and w =, compute (u v) w and u (v w) Are you surprised at the results? 45 Let a and b denote arbitrary scalars Find au + bv given u = and v = 4 Then find the equation of the plane that consists of all such linear combinations of u and v A?
5 Extra Problems 5 46 Find two vectors perpendicular to both u = and v = 47 Find a vector of length 5 that is perpendicular to both u = 48 Verify that u v = u v sin θ with u = and v = and v = 5 49 Find the equation of the plane passing through A, B, and C, given A(,,), B(,,5), C(,, 4) x 5 Find the equation of the plane containing P(,, ) and the line with equation y = z 4 + t 5 Find equations of [ the form [ ax [ + by = c for the lines in the Euclidean plane with the x vector equation = + t y 5 Find vector equations for the line in the Euclidean plane whose equation is y = x 5 The intersection of two planes is, in general, a line (a) Find three points on the line that is the intersection of the planes with equations x + y + z = 5 and x y + z = (b) Find a vector equation for the line in 5(a) x 54 Determine, with a simple reason, whether the line with equation y = +t 5 z and the plane with the equation x y + z = 4 intersect Find any points of intersection 55 Let π be the plane with equation x y + 5z = Find a vector perpendicular to π 56 Let A(,, ), B(,, ), C(, 5,), D(, 4,) be the indicated points of  space (a) Show that A,B,C,D are the vertices of a parallelogram and find the area of this figure (b) Find the area of triangle ABC
6 6 Extra Problems 57 Find the area of the triangle in the xyplane with vertices A(, ), B(4, 4), C(5, ) 58 Suppose a determinant a b [ [ c d = Show that one of the vectors c a, is a d b scalar multiple of the other There are four cases to consider, depending on whether or not a = or b = x x 59 Determine whether or not the lines with equations y = + t and y = z z 4 +t intersect Find any point of intersection 5 6 Let l and l be the lines with the equations x l : y = + t ; l : z x y = + t z a b Given that these lines intersect at right angles, find the values of a and b 6 Find the equation of the plane each point of which is equidistant from the points P(,,) and Q(,, ) 6 When does the projection of u on a nonzero vector v have a direction opposite that of v? 6 Let u = and v = Determine whether w = is in the plane spanned by u and v 64 In each case below, find the projection of u on v and the projection of v on u (a) u = and v = (b) u = and v = 4 65 Find the distance from point P(,,) to the line l with equation t x y = z + 66 Find the distance from P(,,) to the plane π with equation 5x y + z = Then find the point of π closest to P 67 Find two orthogonal vectors that span each of the given planes:
7 Extra Problems 7 (a) the plane spanned by u = and v = 4 (b) the plane with equation x y + 4z = 68 (a) Find two orthogonal vectors in the plane π with equation x y + z = (b) Use your answer to part (a) to find the projection of w = on π 69 (a) What is the projection of w = onto the plane π with equation x+y z =? x (b) What is the projection of w = y onto π? z 7 Find the projection of w = on the plane π whose equation is x + y z = 7 Suppose a vector w is a linear combination of nonzero orthogonal vectors e and f Prove that w = w e e e e+ w f f f f f w f f f f w e e e e e x x 7 (a) The lines with vector equations y = + t and y = z 4 z are parallel Why? (b) Find the distance between the lines in (a) w 4 + t 7 Most theorems in mathematics can be stated in the form If P, then Q, symbolically P = Q (Read implies for = ) The converse of this statement is Q = P: If Q, then P What is the converse of Theorem 5? 74 Don has a theory that to find the projection of a vector w on a plane π (through the origin), one could project w on a normal n to the plane, and take proj π w = w proj n w (a) Is Don s theory correct? Explain by applying the definition of projection onto a plane (b) Illustrate Don s theory using the plane with equation x y+z = and w = 6 6
8 8 Extra Problems 75 (The distance between skew lines) Let l and l be the lines with equations x 4 x l : y = + t, l : y = + t 6 z z 7 (a) Show that l and l are not parallel and that they do not intersect (Such lines are called skew) (b) Find the equation of the plane that contains l and is parallel to l (c) Use the result of (b) to find the (shortest) distance between l and l 76 Let u =, v =, x =, and y = Find each of the following, if possible (a) x y + u (b) y 77 Find a and b, if possible, given a b a b = b 6 b 8 c a 78 Let u = b, v =, x =, and y = a Find a, b, and c, if possible, given u v = y 79 Find a unit vector that has the same direction as u = 8 Let u = and v = Find all scalars c with the property that cu = 8 8 Find the angle (to the nearest tenth of a radian and to the nearest degree) between u = and v = 8 Suppose u is a vector of length 6 and v is a unit vector orthogonal to u + v Find u v 8 Suppose u, v, x, and y are unit vectors such that v x =, u and y are orthogonal, the angle between v and y is 45, and the angle between u and x is 6 Find (u+v) (x y)
9 Extra Problems 9 84 Determine, with justification, whether v = is a linear combination of v 4 = 5, v = 4, v =, v 5 4 =, and v 5 = Verify the Cauchy Schwarz and triangle inequalities for u and v = 86 Determine whether v =, v =, and v =, v 4 = are linearly 4 4 independent or linearly dependent 87 Why have generations of linear algebra students refused to memorize the definitions of linear independence and linear dependence? 88 Suppose you were asked to prove that vectors v,v,,v k are linearly independent What would be the first line of your proof? What would be the last line? x y 89 Find x and y, if possible, given x y x + y y x + y = x + y [ [ a + x 9 Find x, y, a, and b if possible, given = y + b 9 Let A = [ x y [, B = z 4 5 Find x, y, and z, if possible, so that (a) A B = C (b) AB = C [ 7, C = 8 a x a + x [ 7, and D = 9 What is the matrix A for which a ij = j? [ 9 Find A+B, A B, and A B given A = [ 94 Find AB and BA (if defined) given A = [ and B = and B = 4 [ 4 6
10 Extra Problems [ x u a b c 95 Let A = and B = y v What are the (,) and (,) entries of d e f z w AB? [ [ [ 96 Write + 4 in the form Ax, for a suitable matrix A and vector x [This 8 question tests understanding of the very important fact expressed in 6 97 Express the system x y = x + 5y = 7 x + y = x 5y = in the form Ax = b 98 The parabola with equation y = ax + bx + c passes through the points (,4) and a (,8) Write a matrix equation whose solution is the vector b c 99 Complete the following sentence: A linear combination of the columns of a matrix A is 4 9 Let A = 4 5 6, x =, and b = Given that Ax = b, write b as a linear combination of the columns of A [ [ Express b = as a linear combination of the columns of A = Then find a vector x so that Ax = b Let I = be the identity matrix Express x = combination of the columns of I 5 as a linear 47 Suppose x, x, and x are threedimensional vectors and A is a matrix such that x x x Ax =, Ax =, Ax = Let X = be the matrix with columns x, x, x What is AX? 4 Let A = and D = explain your reasoning Find AD without calculation and
11 Extra Problems 5 Let A = [ 4 [ 4, B = 4 [, C = 5 Compute AB, AC, AB + AC, B + C, and A(B + C) 6 Let A be a square matrix Does the equation (A [ I)(A + I) = imply A = I or A = I? Answer by considering the matrix A = 7 If A is a matrix and x is a vector such that x+ax is a vector, what can you say about the size of A (and why)? 8 If A and B are m n matrices and Ax = Bx for each vector x in R n, show that A = B 9 Let x = x x x n be nonzero and let y be any vector in Rm Prove that there exists an m n matrix A such that Ax = y Determine whether or not A = are inverses Given A = Let A = [ 5 [ 7 9 (a) Compute AB and BA and B = 5 5 find A and use this matrix to solve the system and B = (b) Is A invertible? Explain [ Let A = and B = A T B T, and B T A T [ 4 4 x + 5y = x y = 7 Compute A T, B T, AB, BA, (AB) T, (BA) T, 4 Verify Property 4 of the transpose of a matrix by showing that if A is an invertible matrix, then so is A T, and (A T ) = (A ) T 5 Suppose A and B are square matrices which commute Prove that (AB) T = A T B T 6 Let A, B, and C be matrices with C invertible If AC = BC, show that A = B 7 Let A be any m n matrix Let x be a vector in R n and let y be a vector in R m Explain the equation y (Ax) = y T Ax
12 Extra Problems 8 Suppose A and B are invertible matrices and X is a matrix such that AXB = A+B What is X? [ 9 Is in row echelon form? Reduce each of the following matrices to row echelon form In each case, identify the pivots and the pivot columns 4 5 (a) 5 (b) Repeat Exercise for the matrix Identify the pivot columns, but this time don t go to the trouble of identifying the pivots x Given A = 4 6 6, x = x and b = 4, solve Ax = b expressing your solution as a vector or as a linear combination of vectors as 6 6 x appropriate Write the system x + x + 5x = in the form Ax = b What is A? What 8x + 7x + 9x = 4 is x? What is b? Solve the system, expressing your answer as a vector or a linear combination of vectors as appropriate 4 In solving the system Ax = b, Gaussian elimination on the augmented matrix [A b led to the row echelon matrix [ 5 The variables were x,x,x,x 4 (a) Circle the pivots Identify the free variables (b) Write the solution to the given system as a vector or as a linear combination of vectors, as appropriate 5 Here is a matrix in row echelon form: It represents the final row echelon matrix after Gaussian elimination was applied to the matrix [A b in an attempt to solve Ax = b If possible, find the solution (in vector
13 Extra Problems form) of Ax = b If there is a solution, state whether this is unique or whether there are infinitely many solutions The vector x is 6 Solve each of the following systems of linear equations by Gaussian elimination and back substitution Write your answers as vectors or as linear combinations of vectors if appropriate (a) x y + z = 4 x + y = x + y 6z = 5 (b) x y + z = 4 (c) x y + z = x + y 6z = 9 x + y 5z = 4 (d) 6x + 8x 5x 5x 4 = 6x + 7x x 8x 4 = 9 8x + x x 9x 4 = 7 Determine whether 6 is a linear combination of the columns of A = 4 8 Determine whether x x is a linear combination of 9 Let v =, v =, v =, and v 4 = and Determine whether every vector in R is a linear combination of v, v, v, and v 4 Determine whether or not the points (,, ), (,, ), (4, 7, ), (,, 9) are coplanar, that is, lie in a plane If the answer is yes, give an equation of a plane on which the points all lie The planes with equations x + z = 5 and x + y = intersect in a line (a) Why? (b) Find an equation of the line of intersection Suppose we want to find a quadratic polynomial p(x) = a + bx + cx that passes through the points (,), (, ), (5,4)
14 4 Extra Problems (a) Write the system of equations that must be solved in order to determine a, b, and c (b) Find the desired polynomial Consider the system 5x + y = a 5x 6y = b (a) Under what conditions (on a and b), if any, does this system fail to have a solution? (b) Under what conditions, if any, does the system have a unique solution? Find any unique solution that may exist (c) Under what conditions are there infinitely many solutions? Find these solutions if and when they exist 4 (a) Find a condition on a, b, and c that is both necessary and sufficient for the system to have a solution x y = a 5x + y = b x + y = c (b) Could the system in part (a) have infinitely many solutions? Explain x + x + x = 5 Solve x + x = by Gaussian elimination and back substitution Write x x x = your answer as vectors or as linear combinations of vectors as appropriate 6 (a) Express the zero vector = as a nontrivial linear combination of the columns of A = (b) Use part (a) to show that A is not invertible 7 Find conditions on a, b, and c (if any) such that the system has a unique solution; x + y = y + z = x z = ax + by + cz =
15 Extra Problems 5 no solution; an infinite number of solutions 8 Write the solution to x + 5x + 7x = 9x = in the form x = x p + x h where x p is a particular solution and x h is a solution of the corresponding homogeneous system 9 Determine whether or not the vectors v =, v =, v =, v 4 = are linearly independent If they are not, write down a nontrivial linear combination of the vectors which equals the zero vector 4 Let E = 4 and let A be an arbitrary n matrix, n (a) Explain the connection between A and EA (b) What is the name for a matrix such as E? (c) Is E invertible? If so, what is its inverse? 4 Suppose A is a n matrix and we would like to modify A by subtracting the second row from the third Write the elementary matrix E for which EA is A modified as specified 4 Let A = Write an elementary matrix E so that EA = Let E be the elementary matrix that adds 4 times row one to row two and F the elementary matrix that subtracts times row two from row three (a) Find E, F, EF, and FE without calculation (b) Write the inverses of the matrices in part (a) without calculation 44 What number xwill force an interchange of rows when Gaussian elimination is applied 5 to 4 x? 45 Let A = a a a n be an m n matrix with columns the vectors a,a,,a n and let B be the upper triangular matrix B = b b b n b b n How are the columns b nn
16 6 Extra Problems of AB related to the columns of B? 46 Write the inverse of each of the following matrices without calculation Explain your reasoning [ (a) A = (b) A = [ 47 Find the inverse of simply by writing it down No calculation is necessary, but explain your reasoning [ 48 Show that the lower triangular matrix L = is invertible by finding a matrix a M with LM = I, the identity matrix 4 49 Let A be a n matrix, E =, D = and P = (a) How are the rows of EA, of DA and of PA related to the rows of A? (b) Find P [ 5 Ian says has no LU factorization but Lynn disagrees Lynn adds the second row to the first, then subtracts the first from the second, like this [ [ [ = U She gets L as the product of certain elementary matrices Who is right? Explain 5 Write L = as the product of elementary matrices (without calculation) 5 and use this factorization to find L 5 Write U = 5 as the product of elementary matrices (without calculation) and use this factorization to find U [ 5 If possible, find an LU factorization of A = and express L as the 7 product of elementary matrices
17 Extra Problems 7 54 Let A = and v = 5 (a) Find an elementary matrix E and an upper triangular matrix U such that EA = U (b) Find a lower triangular matrix L such that A = LU (c) Express Av as a linear combination of the columns of A 6 55 Let A = 5 Find an LU factorization of A by moving A to a row 4 echelon matrix (with the elementary row operations), keeping track of the elementary matrices Express L as the product of elementary matrices 56 Find, if possible, an LU factorization of each of the following matrices A: [ (a) (b) [ [ [ 57 Let A = = = LU [ [ x (a) Use this factorization of A to solve the system A = y 9 [ (b) Express as a linear combination of the columns of A 9 [ [ [ 6 58 Use the factorization = to solve Ax = b for x = 5 [ [ [ x 6, with b =, where A = x Let A = and b = (a) Solve the system Ax = b by applying the method of Gaussian elimination to the augmented matrix (b) Factor A = LU (c) Why is A invertible? (d) Find the first column of A 6 Solve x + x = 8 4x + 9x = as follows:
18 8 Extra Problems i Write the system in the form Ax = b ii Factor A = LU iii Solve Ly = b for y by forward substitution and Ux = y by back substitution Finally, write the solution to Ax = b in vector form 6 Given A = and b =, i find an LU factorization of A and ii use this to solve Ax = b 6 Find a PLU factorization of A = [ 6 Suppose A and B are symmetric matrices Is AB symmetric (in general)? 64 A matrix A is skewsymmetric if A T = A Show that if A and B are n n skewsymmetric matrices, then so is A + B 65 True or false: The pivots of an invertible matrix A are the nonzero diagonal entries of U in an LU or a PLU factorization 66 If A is a symmetric n n matrix and x is a vector in R n, show that x T A x = Ax [ 67 Determine whether or not A = and B = [ are inverses Find the reduced row echelon form of Determine whether or not each of the following matrices has an inverse Find the inverse whenever this exists [ (a) (b) Consider the system of equations x x + x = x + x + x = x + x = i Write this in the form Ax = b ii Solve the system by finding A
19 Extra Problems 9 iii Express as a linear combination of the vectors,, 7 Given that A is a matrix and 7 Express [ [ A[ 5 as the product of elementary matrices 7 Use Theorem 86 to determine whether or not [ 7 = [ is invertible, find A 74 Given two n n matrices X and Y, how do you determine whether or not Y = X? 75 Is the factorization of an invertible matrix as the product of elementary matrices unique? Explain 76 For each of the following matrices A, i find the matrix M of minors, find the matrix C of cofactors, and compute the products AC T and C T A; ii find deta; iii if A is invertible, find A [ (a) A = Find the determinant of A = row 78 Let A = (a) Find the matrix of minors (b) Find cofactor matrix C (b) A = expanding by cofactors of the third (c) Find the dot product of the second row of A and the third row of C Could this result have been anticipated? Explain (d) Find the dot product of the first column of A and the second column of C Could this result have been anticipated? Explain (e) Find the dot product of the second column of A and the second column of C What is the significance of this number?
20 Extra Problems 79 Let A = of A and suppose C = (a) Find the values of c, c and c (b) Find deta c c c (c) Is A invertible? Explain without further calculation (d) Find A, if this exists 8 For what values of x are the given matrices singular? [ x x (a) (b) x + 4 x 7 is the matrix of cofactors 8 Professor G was making up a homework problem for his class when he spilled a cup of coffee over his work (This happens to Prof G all too frequently) All that remained legible was this: A = (a) Find the missing entries (b) Find A 8 Find all values of a which make P(a) = 8 If the matrix of cofactors of a matrix A is ; matrix of cofactors = C = a singular a a + a [ Given det A = 5 and the matrix of cofactors C =, find A [ 9 4 5, find the matrix A 85 Given that = 8, find
21 Extra Problems (a) (c) (e) (b) (d) Let A and B be 5 5 matrices with det( A) = 4 and det B = Find deta, detb and det AB Let A = 5 77 Find det A, deta, and deta 5 88 Let A = (a) Find the determinant of A expanding by cofactors of the first column (b) Find the determinant of A by reducing it to an upper triangular matrix [ [ 89 Determine whether the vectors, are linearly independent or linearly dependent 4 9 Find the determinants of each of the following matrices by reducing to upper triangular matrices (a) 6 (b) If P is a permutation matrix and 6 7 PA = = = LU,
22 Extra Problems what are the possible values for deta? 9 Give at least three conditions on the rows of a matrix A that guarantee that A is singular 9 Suppose each column of a matrix A is a linear combination of two vectors u and v Find an argument involving the determinant that shows that A is singular 94 Find the following determinants by methods of your choice (a) (b) 4 4 (c) a b c a g d 95 Suppose d e f = Find b h e g h i c i f 96 Suppose A is a square matrix Explain how an LDU factorization of A could prove 4 helpful in finding the determinant of A and illustrate with A = Suppose A is an n n invertible matrix and A = (I A), where I is the n n identity matrix Show that det A is a power of 98 Let A = 5 4 and v = 4 Is v an eigenvector for A? Explain and, if your answer is yes, give the corresponding eigenvalue [ [ [ [ 99 Given A =, determine whether or not each of v 4 =, v =, v =, [ [ v 4 =, v 5 = is an eigenvector of A Justify your answers and state the eigenvalues which correspond to any eigenvectors In each of the following situations, determine whether or not each of the given scalars is an eigenvalue of the given matrix Justify your answers [ (a) A = ;,,,,5 (b) A = 4 4 ;,,4,5,6 4 4
23 Extra Problems Let A = [ (a) Show that [ is an eigenvector of A and state the corresponding eigenvalue (b) What is the characteristic polynomial of A? (c) What are the eigenvalues of A? Find the characteristic polynomial, the (real) eigenvalues, and the corresponding eigenspaces of each of the following matrices A [ 5 8 (a) (b) [ Let A =, let P be the point (x,y) in the Cartesian plane, and let v be the vector OP Let Q be the point such that Av = OQ (a) Find the coordinates of Q (b) Show that the line l with equation y = x is the right bisector of the line PQ joining P to Q; that is, PQ is perpendicular to l and the lines intersect at the midpoint of PQ (c) Describe the action of multiplication of A in geometrical terms (d) Find the eigenvalues and corresponding eigenspaces of A without calculation [ 4 Multiplication by A = is reflection in a line Find the equation of this line Use the fact that the matrix is a reflection to find all eigenvectors and eigenvalues 5 Let λ be an eigenvalue of an n n matrix A and let u,v be eigenvectors corresponding to λ If a and b are scalars, show that A(au + bv) = λ(au + bv) Thus, if au + bv, it is also an eigenvector of A corresponding to λ 6 Suppose the components of each row of an n n matrix A add to 7 Prove that 7 is an eigenvalue of A 7 Suppose A, B, and P are n n matrices with P invertible and PA = BP Show that every eigenvalue of A is an eigenvalue of B 8 What are the eigenvalues of? What are the corresponding eigenvalues?
24 4 Extra Problems 9 Let v be an eigenvector of a matrix A with corresponding eigenvalue λ Is v an eigenvector of 5A? If so, what is the corresponding eigenvalue? Explain (a) Suppose a matrix A is similar to the identity matrix What can you say about A? Explain [ (b) Can be similar to the identity matrix? Explain [ (c) Is A = diagonalizable? Explain Suppose A, B, and C are n n matrices If A is similar to B and B is similar to C, show that A is similar to C [ [ Use the given fact that A = and B = are similar matrices to 4 8 find the determinant of B and the characteristic polynomial of B [ (a) Find all eigenvalues and eigenvectors of A = (b) Explain why the results of part (a) show that A is not diagonalizable [ 4 Let A = (a) Show that λ = is an eigenvalue of A and find the corresponding eigenspace (b) Given that the eigenvalues of A are and, what is the characteristic polynomial of A? (c) Why is A diagonalizable? [ (d) Given that is an eigenvector of A corresponding to λ =, find a matrix P [ such that P AP = [ 6 (e) Let Q = Find Q 4 AQ without calculation 5 Let A = (a) What is the characteristic polynomial of A? (b) Why is A diagonalizable? (c) Find a matrix P such that P AP = 4
25 Extra Problems 5 6 Determine whether or not each of the matrices A given below is diagonalizable If it is, find an invertible matrix P and a diagonal matrix D such that P AP = D If it isn t, explain why (a) A = 7 Let A = [ [ (b) A = (c) A = (a) Find an invertible matrix P and a diagonal matrix D such that P AP = D (b) Find a matrix B such that B = A [Hint: First, find a matrix D such that D = D 8 One of the most important and useful theorems in linear algebra says that a real symmetric matrix A is always diagonalizable in fact, orthogonally diagonalizable; that is, there is a matrix P with orthogonal columns such that P AP is a diagonal matrix) Illustrate this fact with respect to the matrix A = [ 9 Repeat Exercise 8 for A = 5 5 [ Prove Theorem in general; that is, if x,x,,x k are eigenvectors of a matrix A whose corresponding eigenvalues λ,λ,,λ k are all different, then x,x,,x k are linearly independent [Hint: If the result is false, then there are scalars c,c,,c k, not all, such that c x + c x + + c k x k = Omitting the terms with c i = and renaming the terms that remain, we may assume that c x + c x + + c l x l = ( ) for some l, where c, c,, c l Among all such equations, suppose the one exhibited is shortest; that is, whenever we have an equation like ( ) (and no coefficients ), there are at least l terms Note that l >, since otherwise c x = and x gives c =, a contradiction Now imitate the proof of Theorem Given that a matrix A is similar to the matrix, what s the characteristic polynomial of A? Explain 4 5 If A is a diagonalizable matrix with all eigenvalues nonnegative, show that A has a square root; that is, A = B for some matrix B [Hint: Exercise 7
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More information4.1 Distance and Length
Chapter Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at vectors
More informationHOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)
HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe
More informationVector Geometry. Chapter 5
Chapter 5 Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at
More informationMath Linear Algebra Final Exam Review Sheet
Math 151 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a componentwise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More informationMATH 369 Linear Algebra
Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine
More informationchapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?
chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "
More informationCHAPTER 8: Matrices and Determinants
(Exercises for Chapter 8: Matrices and Determinants) E.8.1 CHAPTER 8: Matrices and Determinants (A) means refer to Part A, (B) means refer to Part B, etc. Most of these exercises can be done without a
More informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationftuiowamath2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST
me me ftuiowamath255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following
More informationELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices
ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS KOLMAN & HILL NOTES BY OTTO MUTZBAUER 11 Systems of Linear Equations 1 Linear Equations and Matrices Numbers in our context are either real numbers or complex
More informationEigenvalues and Eigenvectors
CHAPTER Eigenvalues and Eigenvectors CHAPTER CONTENTS. Eigenvalues and Eigenvectors 9. Diagonalization. Complex Vector Spaces.4 Differential Equations 6. Dynamical Systems and Markov Chains INTRODUCTION
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationand let s calculate the image of some vectors under the transformation T.
Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationMATH 1210 Assignment 4 Solutions 16RT1
MATH 1210 Assignment 4 Solutions 16RT1 Attempt all questions and show all your work. Due November 13, 2015. 1. Prove using mathematical induction that for any n 2, and collection of n m m matrices A 1,
More information4 Elementary matrices, continued
4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. To repeat the recipe: These matrices are constructed by performing the given row
More informationMATH10212 Linear Algebra B Homework Week 5
MATH Linear Algebra B Homework Week 5 Students are strongly advised to acquire a copy of the Textbook: D C Lay Linear Algebra its Applications Pearson 6 (or other editions) Normally homework assignments
More informationChapter 1: Systems of Linear Equations and Matrices
: Systems of Linear Equations and Matrices Multiple Choice Questions. Which of the following equations is linear? (A) x + 3x 3 + 4x 4 3 = 5 (B) 3x x + x 3 = 5 (C) 5x + 5 x x 3 = x + cos (x ) + 4x 3 = 7.
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More information18.06SC Final Exam Solutions
18.06SC Final Exam Solutions 1 (4+7=11 pts.) Suppose A is 3 by 4, and Ax = 0 has exactly 2 special solutions: 1 2 x 1 = 1 and x 2 = 1 1 0 0 1 (a) Remembering that A is 3 by 4, find its row reduced echelon
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342  Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More information1. Select the unique answer (choice) for each problem. Write only the answer.
MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More informationMath 215 HW #9 Solutions
Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith
More informationLecture 23: 6.1 Inner Products
Lecture 23: 6.1 Inner Products WeiTa Chu 2008/12/17 Definition An inner product on a real vector space V is a function that associates a real number u, vwith each pair of vectors u and v in V in such
More informationTopic 15 Notes Jeremy Orloff
Topic 5 Notes Jeremy Orloff 5 Transpose, Inverse, Determinant 5. Goals. Know the definition and be able to compute the inverse of any square matrix using row operations. 2. Know the properties of inverses.
More informationA matrix is a rectangular array of. objects arranged in rows and columns. The objects are called the entries. is called the size of the matrix, and
Section 5.5. Matrices and Vectors A matrix is a rectangular array of objects arranged in rows and columns. The objects are called the entries. A matrix with m rows and n columns is called an m n matrix.
More informationChapter 2: Matrices and Linear Systems
Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers
More informationLinear Algebra. Workbook
Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx
More informationTopic 2 Quiz 2. choice C implies B and B implies C. correctchoice C implies B, but B does not imply C
Topic 1 Quiz 1 text A reduced rowechelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correctchoice may have 0, 1, 2, or 3 choice may have 0,
More informationEigenvalues and Eigenvectors
Chapter 6 Eigenvalues and Eigenvectors 6. Introduction to Eigenvalues Eigenvalues are the key to a system of n differential equations : dy=dt ay becomes dy=dt Ay. Now A is a matrix and y is a vector.y.t/;
More informationMath 313 (Linear Algebra) Exam 2  Practice Exam
Name: Student ID: Section: Instructor: Math 313 (Linear Algebra) Exam 2  Practice Exam Instructions: For questions which require a written answer, show all your work. Full credit will be given only if
More informationMath 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name:
Math 34/84  Exam Blue Exam Solutions December 4, 8 Instructor: Dr. S. Cooper Name: Read each question carefully. Be sure to show all of your work and not just your final conclusion. You may not use your
More information2.3. VECTOR SPACES 25
2.3. VECTOR SPACES 25 2.3 Vector Spaces MATH 294 FALL 982 PRELIM # 3a 2.3. Let C[, ] denote the space of continuous functions defined on the interval [,] (i.e. f(x) is a member of C[, ] if f(x) is continuous
More informationMATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants.
MATH 323 Linear Algebra Lecture 6: Matrix algebra (continued). Determinants. Elementary matrices Theorem 1 Any elementary row operation σ on matrices with n rows can be simulated as left multiplication
More informationMath 110 Linear Algebra Midterm 2 Review October 28, 2017
Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections
More informationBASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x
BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,
More informationMATH10212 Linear Algebra B Homework 7
MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments
More information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra  Test File  Spring Test # For problems  consider the following system of equations. x + y  z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationSolutions to Math 51 First Exam April 21, 2011
Solutions to Math 5 First Exam April,. ( points) (a) Give the precise definition of a (linear) subspace V of R n. (4 points) A linear subspace V of R n is a subset V R n which satisfies V. If x, y V then
More informationSystems of Linear Equations. By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University
Systems of Linear Equations By: Tri Atmojo Kusmayadi and Mardiyana Mathematics Education Sebelas Maret University Standard of Competency: Understanding the properties of systems of linear equations, matrices,
More informationMathematical Foundations
Chapter 1 Mathematical Foundations 1.1 BigO Notations In the description of algorithmic complexity, we often have to use the order notations, often in terms of big O and small o. Loosely speaking, for
More informationMath 20F Practice Final Solutions. Jorel Briones
Math 2F Practice Final Solutions Jorel Briones Jorel Briones / Math 2F Practice Problems for Final Page 2 of 6 NOTE: For the solutions to these problems, I skip all the row reduction calculations. Please
More information18.06 Problem Set 2 Solution
18.06 Problem Set 2 Solution Total: 100 points Section 2.5. Problem 24: Use GaussJordan elimination on [U I] to find the upper triangular U 1 : 1 a b 1 0 UU 1 = I 0 1 c x 1 x 2 x 3 = 0 1 0. 0 0 1 0 0
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More information(c)
1. Find the reduced echelon form of the matrix 1 1 5 1 8 5. 1 1 1 (a) 3 1 3 0 1 3 1 (b) 0 0 1 (c) 3 0 0 1 0 (d) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (e) 1 0 5 0 0 1 3 0 0 0 0 Solution. 1 1 1 1 1 1 1 1
More informationLinear Equations and Matrix
1/60 ChiaPing Chen Professor Department of Computer Science and Engineering National Sun Yatsen University Linear Algebra Gaussian Elimination 2/60 Alpha Go Linear algebra begins with a system of linear
More informationAppendix C Vector and matrix algebra
Appendix C Vector and matrix algebra Concepts Scalars Vectors, rows and columns, matrices Adding and subtracting vectors and matrices Multiplying them by scalars Products of vectors and matrices, scalar
More informationChapter 3: Theory Review: Solutions Math 308 F Spring 2015
Chapter : Theory Review: Solutions Math 08 F Spring 05. What two properties must a function T : R m R n satisfy to be a linear transformation? (a) For all vectors u and v in R m, T (u + v) T (u) + T (v)
More information80 min. 65 points in total. The raw score will be normalized according to the course policy to count into the final score.
This is a closed book, closed notes exam You need to justify every one of your answers unless you are asked not to do so Completely correct answers given without justification will receive little credit
More informationChapter Two Elements of Linear Algebra
Chapter Two Elements of Linear Algebra Previously, in chapter one, we have considered single first order differential equations involving a single unknown function. In the next chapter we will begin to
More informationMATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.
MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of
More informationMATH 300, Second Exam REVIEW SOLUTIONS. NOTE: You may use a calculator for this exam You only need something that will perform basic arithmetic.
MATH 300, Second Exam REVIEW SOLUTIONS NOTE: You may use a calculator for this exam You only need something that will perform basic arithmetic. [ ] [ ] 2 2. Let u = and v =, Let S be the parallelegram
More informationI. Multiple Choice Questions (Answer any eight)
Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam  Course Instructor : Prashanth L.A. Date : Sep24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY
More informationMATH10212 Linear Algebra B Homework Week 3. Be prepared to answer the following oral questions if asked in the supervision class
MATH10212 Linear Algebra B Homework Week Students are strongly advised to acquire a copy of the Textbook: D. C. Lay Linear Algebra its Applications. Pearson, 2006. ISBN 052128714. Normally, homework
More information18.06 Problem Set 8  Solutions Due Wednesday, 14 November 2007 at 4 pm in
806 Problem Set 8  Solutions Due Wednesday, 4 November 2007 at 4 pm in 206 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state
More informationMTH 102A  Linear Algebra II Semester
MTH 0A  Linear Algebra  056II Semester Arbind Kumar Lal P Field A field F is a set from which we choose our coefficients and scalars Expected properties are ) a+b and a b should be defined in it )
More informationPrepared by: M. S. KumarSwamy, TGT(Maths) Page
Prepared by: M S KumarSwamy, TGT(Maths) Page  119  CHAPTER 10: VECTOR ALGEBRA QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 06 marks Vector The line l to the line segment AB, then a
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
More informationCHAPTER 6. Direct Methods for Solving Linear Systems
CHAPTER 6 Direct Methods for Solving Linear Systems. Introduction A direct method for approximating the solution of a system of n linear equations in n unknowns is one that gives the exact solution to
More informationProblem Set # 1 Solution, 18.06
Problem Set # 1 Solution, 1.06 For grading: Each problem worths 10 points, and there is points of extra credit in problem. The total maximum is 100. 1. (10pts) In Lecture 1, Prof. Strang drew the cone
More informationCreated by T. Madas VECTOR PRACTICE Part B Created by T. Madas
VECTOR PRACTICE Part B THE CROSS PRODUCT Question 1 Find in each of the following cases a) a = 2i + 5j + k and b = 3i j b) a = i + 2j + k and b = 3i j k c) a = 3i j 2k and b = i + 3j + k d) a = 7i + j
More informationLinear Algebra Using MATLAB
Linear Algebra Using MATLAB MATH 5331 1 May 12, 2010 1 Selected material from the text Linear Algebra and Differential Equations Using MATLAB by Martin Golubitsky and Michael Dellnitz Contents 1 Preliminaries
More informationINVERSE OF A MATRIX [2.2]
INVERSE OF A MATRIX [2.2] The inverse of a matrix: Introduction We have a mapping from R n to R n represented by a matrix A. Can we invert this mapping? i.e. can we find a matrix (call it B for now) such
More information5.3.5 The eigenvalues are 3, 2, 3 (i.e., the diagonal entries of D) with corresponding eigenvalues. Null(A 3I) = Null( ), 0 0
535 The eigenvalues are 3,, 3 (ie, the diagonal entries of D) with corresponding eigenvalues,, 538 The matrix is upper triangular so the eigenvalues are simply the diagonal entries, namely 3, 3 The corresponding
More informationMath Camp Notes: Linear Algebra I
Math Camp Notes: Linear Algebra I Basic Matrix Operations and Properties Consider two n m matrices: a a m A = a n a nm Then the basic matrix operations are as follows: a + b a m + b m A + B = a n + b n
More informationav 1 x 2 + 4y 2 + xy + 4z 2 = 16.
74 85 Eigenanalysis The subject of eigenanalysis seeks to find a coordinate system, in which the solution to an applied problem has a simple expression Therefore, eigenanalysis might be called the method
More informationFall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop
Fall 2017 Inverse of a matrix Authors: Alexander Knop Institute: UC San Diego RowColumn Rule If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding
More informationMTH50 Spring 07 HW Assignment 7 {From [FIS0]}: Sec 44 #4a h 6; Sec 5 #ad ac 4ae 4 7 The due date for this assignment is 04/05/7 Sec 44 #4a h Evaluate the erminant of the following matrices by any legitimate
More informationMA 242 LINEAR ALGEBRA C1, Solutions to First Midterm Exam
MA 242 LINEAR ALGEBRA C Solutions to First Midterm Exam Prof Nikola Popovic October 2 9:am  :am Problem ( points) Determine h and k such that the solution set of x + = k 4x + h = 8 (a) is empty (b) contains
More informationMATH10212 Linear Algebra B Homework Week 4
MATH22 Linear Algebra B Homework Week 4 Students are strongly advised to acquire a copy of the Textbook: D. C. Lay Linear Algebra and its Applications. Pearson, 26. ISBN 5228734. Normally, homework
More informationMidterm 2 Solutions, MATH 54, Linear Algebra and Differential Equations, Fall 2014
Name (Last, First): Student ID: Circle your section: 2 Shin 8am 7 Evans 22 Lim pm 35 Etcheverry 22 Cho 8am 75 Evans 23 Tanzer 2pm 35 Evans 23 Shin 9am 5 Latimer 24 Moody 2pm 8 Evans 24 Cho 9am 254 Sutardja
More information(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.
1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III
More informationLinear Algebra: Sample Questions for Exam 2
Linear Algebra: Sample Questions for Exam 2 Instructions: This is not a comprehensive review: there are concepts you need to know that are not included. Be sure you study all the sections of the book and
More informationUMA Putnam Talk LINEAR ALGEBRA TRICKS FOR THE PUTNAM
UMA Putnam Talk LINEAR ALGEBRA TRICKS FOR THE PUTNAM YUFEI ZHAO In this talk, I want give some examples to show you some linear algebra tricks for the Putnam. Many of you probably did math contests in
More informationEigenvalues by row operations
Eigenvalues by row operations Barton L. Willis Department of Mathematics University of Nebraska at Kearney Kearney, Nebraska 68849 May, 5 Introduction There is a wonderful article, Down with Determinants!,
More informationVectors and Plane Geometry
Vectors and Plane Geometry Karl Heinz Dovermann Professor of Mathematics University of Hawaii January 7, 0 Preface During the first week of the semester it is difficult to get started with the course
More informationProblem Set 1. Homeworks will graded based on content and clarity. Please show your work clearly for full credit.
CSE 151: Introduction to Machine Learning Winter 2017 Problem Set 1 Instructor: Kamalika Chaudhuri Due on: Jan 28 Instructions This is a 40 point homework Homeworks will graded based on content and clarity
More information( ) = ( ) ( ) = ( ) = + = = = ( ) Therefore: , where t. Note: If we start with the condition BM = tab, we will have BM = ( x + 2, y + 3, z 5)
Chapter Exercise a) AB OB OA ( xb xa, yb ya, zb za),,, 0, b) AB OB OA ( xb xa, yb ya, zb za) ( ), ( ),, 0, c) AB OB OA x x, y y, z z (, ( ), ) (,, ) ( ) B A B A B A ( ) d) AB OB OA ( xb xa, yb ya, zb za)
More informationFundamentals of Linear Algebra. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Fundamentals of Linear Algebra Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 PREFACE Linear algebra has evolved as a branch of mathematics with wide range of applications to the natural
More informationSolutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015
Solutions to Final Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated /5/05 Answers This page contains answers only. See the following pages for detailed solutions. (. (a x. See
More informationCS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra 1/33
Linear Algebra 1/33 Vectors A vector is a magnitude and a direction Magnitude = v Direction Also known as norm, length Represented by unit vectors (vectors with a length of 1 that point along distinct
More informationELEMENTARY LINEAR ALGEBRA
ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 7th April 013 Comments to the author at keithmatt@gmail.com Chapter 1 LINEAR EQUATIONS 1.1
More informationCHAPTER 1 MATRICES SECTION 1.1 MATRIX ALGEBRA. matrices Configurations like π 6 1/2
page 1 of Section 11 CHAPTER 1 MATRICES SECTION 11 MATRIX ALGEBRA matrices Configurations like 2 3 4 1 6 5, 2 3 7 1 2 π 6 1/2 are called matrices The numbers inside the matrix are called entries If the
More informationLecture 11: Eigenvalues and Eigenvectors
Lecture : Eigenvalues and Eigenvectors De nition.. Let A be a square matrix (or linear transformation). A number λ is called an eigenvalue of A if there exists a nonzero vector u such that A u λ u. ()
More informationMaths for Signals and Systems Linear Algebra in Engineering
Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE
More informationSOLUTIONS: ASSIGNMENT Use Gaussian elimination to find the determinant of the matrix. = det. = det = 1 ( 2) 3 6 = 36. v 4.
SOLUTIONS: ASSIGNMENT 9 66 Use Gaussian elimination to find the determinant of the matrix det 1 1 4 4 1 1 1 1 8 8 = det = det 0 7 9 0 0 0 6 = 1 ( ) 3 6 = 36 = det = det 0 0 6 1 0 0 0 6 61 Consider a 4
More information(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =
. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)
More informationNotes on Mathematics
Notes on Mathematics  12 1 Peeyush Chandra, A. K. Lal, V. Raghavendra, G. Santhanam 1 Supported by a grant from MHRD 2 Contents I Linear Algebra 7 1 Matrices 9 1.1 Definition of a Matrix......................................
More informationLinear Algebra & Geometry why is linear algebra useful in computer vision?
Linear Algebra & Geometry why is linear algebra useful in computer vision? References: Any book on linear algebra! [HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia
More informationChapter 8. Rigid transformations
Chapter 8. Rigid transformations We are about to start drawing figures in 3D. There are no builtin routines for this purpose in PostScript, and we shall have to start more or less from scratch in extending
More information4.3  Linear Combinations and Independence of Vectors
 Linear Combinations and Independence of Vectors De nitions, Theorems, and Examples De nition 1 A vector v in a vector space V is called a linear combination of the vectors u 1, u,,u k in V if v can be
More informationSolutions to Midterm 2 Practice Problems Written by Victoria Kala Last updated 11/10/2015
Solutions to Midterm 2 Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated //25 Answers This page contains answers only. Detailed solutions are on the following pages. 2 7. (a)
More information