# MATH 369 Linear Algebra

Size: px
Start display at page:

## Transcription

1 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 a, b and c such that the parabola y = ax 2 + bx + c passes through the points A, B and C. a) A(2, ), B( 3, ), C(, 0). b) A(2, 5), B(3, 0), C(4, 9). Problem # 3 Find the integer coefficients a, b, c, d so that the following Chemical reaction formula makes sense: ac 2 H 6 + bo 2 cco 2 + dh 2 O. Problem # 4 Let A = 0 2. Find A. 2 4 Problem # 5 Solve each of the following systems: a) x + x 2 2x 3 +4x 4 =5 2x +2x 2 3x 3 + x 4 =3 3x +3x 2 4x 3 2x 4 = b) x + x 2 2x 3 +3x 4 =4 2x +3x 2 +3x 3 x 4 =3 5x +7x 2 +4x 3 + x 4 =5 Problem # 6 Write each of the following matrices as a product of elementary matrices: [ ] 3 a) b) c) Problem # 7 Page 29: 20c). Show your work for full credit.

2 Assignment # 2 Problem # 8 Let A = 0 2. Find A. 2 4 Problem # 9 Write each of the following matrices as a product of elementary matrices: [ ] 3 a) b) c) Problem # 0 Page 27: 9). Show your work for full credit. Problem # Solve each of the following systems. Write down the solution set in the form (special solution plus basic solutions). a) 2x 3x 2 6x 3 5x 4 +2x 5 =7 x 3 +3x 4 7x 5 =6 x 4 2x 5 = b) 2x 6y+7z= 4y+3z=8 +2z=4 c) x+2y 3z=2 2x+3y+ z=4 3x+4y+5z=8 Problem # 2 Compute the partial fraction decomposition of the function f = 3x2 + 7x 2 (x + )(x 2 4).

3 Assignment # 3 Problem # 3 Solve the following system. solution): Write down the solution set in the form (distinguished solution plus basic x +3x 2 2x 3 + 5x 4 =4 2x +8x 2 x 3 + 9x 4 =9 3x +5x 2 2x 3 +7x 4 =7 Problem # 4 Let A = 5. Find A Problem # 5 Let A = Find the LU decomposition of A Problem # 6 The determinant of a 3 3 matrix is A = a, a,2 a,3 a 2, a 2,2 a 2,3 a 3, a 3,2 a 3,3 a, a 2,2 a 3,3 + a,2 a 2,3 a 3, + a,3 a 2, a 3,2 a, a 3,2 a 2,3 a 2, a,2 a 3,3 a 3, a 2,2 a,3. compute the determinant of the matrix in Problem 5. Then compute the product of the diagonal elements in the LU-decomposition (i.e., both L and U). Do you notice anything special? Problem # 7 The vector is in the set. What value of the parameters produces that vector? 2 3 a) 2, i + 0 j i, j R 0 b) 0 4, m n m, n R 2 0

4 Assignment # 4 Problem # 8 Is W a subspace of V or not? V W Yes No R 3 all (a, b, c), a 0 R 3 all (a, b, c), a = c + b R 3 all (a, b, c), a 2 + b 2 + c 2 real polynomials polynomials with integer coefficients real polynomials real polynomials, coeff. of x 2 is real polynomials odd polynomials sequences in R polynomials with coefficients in R 2 2 matrices 2 2 diagonal matrices 3 3 matrices 3 3 upper triangular matrices 2 2 matrices 2 2 symmetric matrices 2 2 matrices singular 2 2 matrices 2 2 matrices nonsingular 2 2 matrices [ ] [ ] a b 0 b all, a, b R all, b R [ b a ] b 0 a b all, a, b, c, d R ad bc = [ c d ] a b all, a, b, c, d R a = d and b = c c d R, scalars = Q all a + b 2, a, b Q R, scalars = Q all a 2, a Q Functions from R to R all a sin(x) + b cos(x), a, b R Problem # 9 Section 3.2 (pg 33) Problem # 20 Linearly dependent or independent? Set of vectors Dependent Independent (, 2), (3, 5) (0, 0, 0) 2t 2 + 4t 3, 4t 2 + 8t 6 (polynomials in t) (,, 2), (2, 3, ), (4, 5, 5) Section 3.3 (pg 44) 2a Section 3.3 (pg 44) 2b

5 Assignment # 5 Problem # 2 Let V be the vector space of real n n matrices. Let U = {A V A = A}, and W = {A V A = A}. a) Show that V = U + W. b) Show that V = U W. Hint: For a given matrix A, consider (A + A )/2 and (A A )/2. Problem # 22 Let V be the set a) Show that V is a subspace of P 5. b) Find a basis for V. Problem # 23 Let { p(x) = 4 i=0 } a i x i P 5, with p() = x =, x 8 2 = 5 6, x 4 3 = 3 5, x 4 = 2, x 5 5 = Find a basis for Span (x, x 2, x 3, x 4, x 5 ). Once you have found a basis {b, b 2,..., b k }, form a matrix whose rows are the b i and compute the row-reduced echelon form. Problem # 24 Let 0 x = 2, x 3 2 = 3 3, x 3 3 = Find a matrix A such that N(A) = Span (x, x 2, x 3 ). Once you have found such a matrix A, compute its row-reduced echelon form.

6 Assignment # 6 Problem # 25 Find a basis for the solution set of the system x 4x 2 +3x 3 4 x = 0 2x 8x 2 +6x 3 2x 4 = 0 Problem # 26 Find a basis and determine the dimension for each of the following: a) 2 2 matrices, b) symmetric 2 2 matrices, c) symmetric 3 3 matrices, d) C as vector space over R. Problem # 27 Find a basis for each of the following subspaces of P 4, the space of cubic polynomials: a) {p(x) P 4 p(7) = 0} b) {p(x) P 4 p(7) = p(5) = 0} c) {p(x) P 4 p(7) = p(5) = p(3) = 0} d) {p(x) P 4 p(7) = p(5) = p(3) = p() = 0} Problem # 28 Find a vector v that completes the basis for the given space: { ( ) } a), v for R 2. { b), 0 }, v for R 3. { 0 0 } c) x, + x 2, v for P 3 the set of quadratic polynomials. Problem # 29 Find a basis for the row-space of the following matrix: Problem # 30 Find the rank of each matrix: a) b) c)

7 Assignment # 7 Problem # 3 Let f : V W be a linear map. Show that f(0) = 0 (0 the zero vector in the appropriate vector space). Problem # 32 Let B = ( v, v 2, v 3 ) be an ordered basis for R 3 with v = 2, v 2 = 3, v 3 = 4 2, 3 Consider the map f : R 3 R 3 a f( x) = b, c where a, b, c are the coordinates of x with respect to B, that is, x = a v + b v 2 + c v a) Compute f( 2 ). b) Show that f is a linear map. 4 c) If f( y) = 2, what is y? 5 Problem # 33 Let D be the differential operator. a) Compute the matrix of D : P 6 P 6 with respect to the ordered basis B = (, x, x 2, x 3, x 4, x 5 ). b) Compute the matrix of D : V V with respect to the ordered basis B = (e x, e x ). Here, V = Span (B). c) Compute the matrix of D : V V with respect to the ordered basis B = (e x + e x, e x e x ). d) Compute the matrix of D : V V with respect to the ordered basis B = (sin(x), cos(x), sin(2x), cos(2x), sin(3x), cos(3x)). Here, V = Span (B). Problem # 34 Let v R n be a non-zero vector. a) Show that the map f : R n R n x v, f( x) = x 2 v v v is the reflection in the hyperplane whose normal vector is v. (here, x y is the dot product between x and y). Hint: it suffices to show that f( v) = v and that f( y) = y for each vector y in the hyperplane orthogonal to v. b) Compute f f by evaluating f(f( x)). What does this tell you? c) Show that f( x) f( y) = x y. What does this tell you?

8 Assignment # 8 Problem # 35 (pg ) Let f be the operator on P 3 defined by f(p(x)) = xp (x) + p (x). a) Find the matrix A representing f with respect to the basis (, x, x 2 ). b) Find the matrix B representing f with respect to the basis (, x, + x 2 ). c) Find the matrix S such that B = S AS. d) If p(x) = a 0 + a x + a 2 ( + x 2 ), calculate f n (p(x)) (here f n is applying f n times). Hint: Note that A = SBS and that (SBS ) n = SB n S. Problem # 36 (pg ) Let V be the subspace of C[a, b] spanned by (, e x, e x ), and let D be the differential operator on V. a) Find the matrix A representing D with respect to the ordered basis (, e x, e x ). b) Find the matrix B representing D with respect to the ordered basis (, cosh x, sinh x). c) Find the transition matrix S representing the change of coordinates from the ordered basis (, e x, e x ) to the ordered basis ( cosh x, sinh x). Recall that cosh x = 2 (ex + e x ), sinh = 2 (xx e x ). d) Verify that B = S AS. Problem # 37 Consider the linear maps L(x, y, z) = (x + 2y, y + 2z, z + 2x) and M(a, b, c) = (a + c, b + a, b + c). a) Compute the matrix of L b) Compute the matrix of M c) Describe the effect of the map M L (i.e. M(L(x, y, z))) d) Compute the matrix of M L. e) How are the matrices in a), b) and d) related? Problem # 38 Compute the eigenvalues and eigenvectors of the following matrix: ( ) 5 3

9 Assignment # 9 Problem # 39 The populations of U.S. cities, suburbs and nonmetro areas in 2007 are described by the following vector x 0 (in units of one million) x 0 = Let p i,j be the probability that a person in area j will move to area i in any given year (city =, suburbs = 2, nonmetro = 3). Assume the transition matrix P = (p i,j ) is P = Determine the long term predictions for the populations of these regions, assuming no change in their total population. Problem # 40 Determine the characteristic polynomial, eigenvalues and the corresponding eigenspaces of the matrix A = Then find a matrix S such that S AS is a diagonal matrix.. Problem # 4 Compute e A for A = [ ].

10 Assignment # 0 Problem # 42 The data below shows the average temperature, in degree celsius, of the earth s surface from 975 through 2002 (source: Worldwatch Institute). Find the equation of the line that best fits the data points. Plot the data and the line that you found. Show all your work for full credit Problem # 43 Using the inner product on P 3 defined by p, q = 0 p(x)q(x)dx, find an orthonormal basis for P 3 starting with the given basis B where B = (x, x + 2, x 2 ). Problem # 44 Let V = R 2 2 with inner product Let W = {A V A = A}. a) Show that A, B = tr(b A). W = {A V A = A} b) Show that every A in V can be written as the sum of matrices from W and from W.

11 Assignment # Problem # 45 According to Kepler s first law, a comet has an elliptic, parabolic or hyperbolic orbit around the sun (gravitational influences from planets ignored). In suitable polar coordinates, the position (r, φ) of a comet satisfies an equation of the form r = β + e(r cos φ) where β is a constant and e is the eccentricity of the orbit, with 0 < e < for an ellipse, e = for a parabola and e > for a hyperbola. Given the data below, determine the type of orbit, and predict the position of the comet at φ = 4.6 radians. [ φ ] 2.4 r Problem # 46 Find the angle θ between: a) u = (, 3, 5, 4) and v = (2, 43, 4, ) in R 4 b) A = [ ] and B = [ ] where A, B = tr(b A). Problem # 47 Let f(x) = { x if 0 x π x + π if π x < 0 a) Find the Fourier approximation for f of degrees n = 2, 3, 4 and 5. b) Sketch y = f(x) along with the polynomials in a). Problem # 48 Let a) Find S such that S AS is diagonal. b) Find C such that C 3 = A. A = ( )

12 Midterm # Preparation Problem # Are the following vectors dependent or independent? a) 2, 3 2, 2, 3 2 3), 5 8, Problem # 2 Compute a basis for the solution space of the following system 2x +3x 2 +x 3 x 4 +4x 5 = 2 3x +4x 2 +3x 3 +x 4 +2x 5 = Problem # 3 Express the vector v in terms of the basis b, b 2, b 3 of R v =, 2 b =, b 2 = 3, 4 b 3 = 2, 2 3 Problem # 4 Find a basis for Span ( v, v 2, v 3, v 4 ) v =, v 2 = 4, v 3 = 0, v 4 = Problem # 5 Find a matrix A whose nullspace is Span ( v, v 2 ) where v =, v = 2 3. Problem # 6 Let V be the space of polynomials a 0 + a t + a 2 t 2 + a 3 t 3 in P 4 with a 0 + a + a 2 + a 3 = 0. Show that V is ia subspace. Let p(x) = 3 t + 2t 2 4t 3. Extend p to a basis for V

13 Problem # 7 Complete the sentence: A set v, v 2,..., v n is linearly dependent if... Problem # 8 Suppose that V = U + W with V a subspace of R 5, dim U = 3, dim W = 3. What are the possibilities for dim V and dim U W?

14 Practice # Problem # Consider the inner product A, B = tr(b A) on the space of all 2 2 matrices over R. Compute Span ([ 0 ]). Problem # 2 If x 3 =, then equals (A) (cx 2 + bx + a) (C) (cx 2 + bx + a) b c x c a x 2 a b x 2 b c x c a a b = a b c b c a c a b ; (B) (cx2 + bx + a) ; (D) (cx2 + bx + a) x b c c a x 2 a b b c x 2 c a x a b Problem # 3 Let A be an n n matrix with A 2 4A + 5I = 0. Show that n is even. Problem # 4 Compute the determinant n n n n A = n n (n ) 0 Problem # 5 Compute the determinant n n... n n 2 n... n A = n n 3... n. n n... n n

15 Problem # 6 Compute the characteristic polynomial of a a A = a a n Problem # 7 For n N, let and a) Is T n a subspace of R[x] over R? b) Is P n a subspace of R[x] over R? T n = {f R[x] deg(f) = n or f = 0} P n = {f R[x] deg(f) n or f = 0}. Problem # 8 Prove that if the rank of the m n matrix A is m, then the system Ax = b is consistent for all b R m. Problem # 9 Let A be an m n matrix and B be the transpose of A. Let P = BA. Prove that A and P have the same null space and the same rank. Problem # 0 Find the LR factorization of Problem # Let A, B, C, D be n n matrices such that CD = DC and D is invertible, show that [ ] A B det = det(ad BC) C D [ ] I 0 Hint: multiply on the right by for suitable X. X I Problem # 2 Let A be an m n matrix with n > m. Show that det A A = 0. Hint: If f and g are linear maps, show that dim Im(g f) dim Im(f).

16 Practice # 2 Problem # Prove: If A and B are nonsingular n n matrices, then AB is also nonsingular and (AB) = B A. Problem # 2 Let A by an m n matrix. a. Explain why the matrix multiplications A T A and AA T are possible. b. Show that A T A and AA T are symmetric Problem # 3 A matrix A is said to be skew-symmetric if A T diagonal entries must all be 0. Problem # 4 Given A = compute A and use it to: [ a. Find a 2 2 matrix X such that AX = B. ] = A. Show that if a matrix is skew-symmetric then its and B = [ ] b. Find a 2 2 matrix Y such that Y A = B. Problem # 5 Let A = [a, a 2, a 3 ] be a 3 3 matrix blocked off by columns. Suppose that a = 3a 2 2a 3. Will the system Ax = 0 have a nontrivial solution? Is A nonsingular? Explain your answers. Problem # 6 Let U be an n n upper triangular matrix with nonzero diagonal entries. a. Explain why U must be nonsingular. b. Explain why U must be upper triangular. Problem # 7 Let A be a nonsingular n n matrix and let B be an n r matrix. Show that the reduced row echelon form of [A, B] is [I, C]. What is C? Problem # 8 Let A and B be n n matrices and define 2n 2n matrices S and M by [ ] [ ] I A AB 0 S =, M = 0 I B 0

17 Determine the block form of S and use it to compute the block form of the product S MS. Problem # 9 Let A be a matrix of the form where α and β are fixed scalars not both equal to 0. [ α β A = 2α 2β ] a. Explain why the system [ 3 Ax = ] must be inconsistent(i.e., a solution does not exist). b. How can one choose a nonzero vector b so that the system Ax = b will be consistent? Explain. Problem # 0 Find all possible choices of c that would make the following matrix singular. 9 c c 3 Note: You could combine information from reducing the matrix and using determinants. Problem # Let A and B be 3 3 matrices with det(a) = 4 and det(b) = 5. Find the value of (a)det(ab), (b)det(3a), (c)det(2ab), (d)det(a B). Problem # 2 Let A be an n n matrix. Is it possible for A 2 +I = 0 in the case where n is odd? Answer the same question in the case where n is even. Note: This problem is different. Use facts about determinants. Problem # 3 In each of the following answer true if the statement is always true and false otherwise. In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true. In each of the following, assume that all the matrices are n n. a. det(ab) = det(ba) b. det(a + B) = det(a) + det(b) c. det(ca) = c det(a) d. det((ab) T ) = det(a) det(b) e. det(a) = det(b) implies A = B. f. det(a k ) = det(a) k g. A triangular matrix is nonsingular if and only if its diagonal entries are all nonzero. h. If x is a nonzero vector in R n and Ax = 0, then det(a) = 0. i. If A 0 n n, but A k = 0 n n for some positive integer k, then A must be singular.

18 Problem # 4 If A is an n n matrix and B = S AS for some nonsingular matrix S, then det(b) = det(a). Problem # 5 Determine whether the following sets form subspaces of R 3. a. {[x, x 2, x 3 ] T : x + x 3 = } b. {[x, x 2, x 3 ] T : x = x 2 = x 3 } c. {[x, x 2, x 3 ] T : x 3 = x + x 2 } Problem # 6 Determine the row space, column space, null space, and dimensions of each for Problem # 7 Determine whether the following are subspaces of P 4 (polynomials of degree 4). a. The set of polynomials in P 4 of even degree b. The set of all polynomials of degree 3 c. The set of all polynomials p(x) in P 4 such that p(0) = 0. d. The set of all polynomials in P 4 having at least one real root. Problem # 8 Which of the following are spanning sets of R 3? a. {[, 0, 0] T, [0,, ] T, [, 0, ] T } b. {[2,, 2] T, [3, 2, 2] T, [2, 2, 0] T } c. {[,, 3] T, [0, 2, ] T } Problem # 9 Which of the following are spanning sets for P 3? a. {, x 2, x 2 2} b. {2, x +, x 2 3, 5x 4 } Problem # 20 Determine whether the following vectors are linearly independent in R 3. a. b. 2, , 2 2 0

19 Problem # 2 Determine whether the following vectors are linearly independent in R 2 2. [ ] [ ] 0 0 a., 0 0 [ ] [ ] [ ] b.,, Problem # 22 Let x, x 2,..., x k be linearly independent vectors in a vector space V. a. If we add a vector x k+ to the collection, will we still have a linearly independent collection of vectors? Explain. b. If we delete a vector, say x k, from the collection, will we still have a linearly independent collection of vectors? Explain. Problem # 23 Given u = [,, ] T, u 2 = [, 2, 2] T, u 3 = [2, 3, 4] T, find the coordinates in terms of u i, i = : 3, of the following vectors. a) [3, 2, 5] T b) [2, 3, 2] T Problem # 24 Given A = a. Compute the RREF form of A. b. What are the dimensions of the row space, column space, and null space of A? How many free variables are there? c. Define the general solution of Ax = 0. Problem # 25 Let A be a 6 5 matrix. If dim(n(a)) = 2, what are the dimensions of the row and column spaces of A? Problem # 26 Prove that a linear system Ax = b is consistent(i.e., a solution exists) if and only if the rank of [A, b] equals the rank of A. Note: The rank equals the number of pivots when bringing a matrix to RREF. Problem # 27 Let A = a. Factor A into a product SDS.

20 b. Use the factorization to compute A 6. a) Use the factorization to compute A. Problem # 28 Let A be a diagonalizable matrix whose eigenvalues are all either or. Show that A = A. Problem # 29 Let A be a 4 4 matrix and let λ be an eigenvalue of algebraic multiplicity 3. If A λi has rank, what is the geometric multiplicity of λ? Problem # 30 Let A be an n n matrix with an eigenvalue λ of multiplicity n. Show that A is diagonalizable if and only if A = λi. Problem # 3 A nilpotent matrix A n n is such that A 2 = A. What are the eigenvaules of A? What is the algebraic multiplicity of each eigenvalue;i.e., if one of the eigenvalues has algebraic multiplicity m, what is the algebraic multiplicity of the other eigenvalues? Hint: Find a polynomial of degree n such that p(a) = 0 n n. Problem # 32 Suppose the matrix A is diagonalizable (in terms of eigenvectors). How are the eigenvectors corresponding to the eigenvalue λ = 0 related to the null space of A? How is this related to the number of free variables for A? Explain. Problem # 33 If S diagonalizes A with a similarity transformation, show that R = (S ) T diagonalizes A T. Problem # 34 Let y =, and let T be the linear operator on R 3 defined by y 2 0, y T (c y + c 2 y 2 + c 3 y 3 ) = (c + c 2 + c 3 )y + (2c + c 3 )y 2 (2c 2 + c 3 )y 3. a. Find a matrix representing T with respect to the ordered basis {y, y 2, y 3 }. b. For each of the following, write the vector x as a linear combination of y, y 2, y 3 and use the matrix from part(a) to determine T (x). (i)x = [7, 5, 2] T, (ii) x = [3, 2, ] T, (iii) x = [, 2, 3] T. Problem # 35 Let T be the linear transformation mapping P 2, polynomials of degree less than or equal to 2, into R 2 define by [ T (p(x)) = 0 p(x)dx ] p(0) Find a matrix A such that [ α T (α + βx) = A β ]

21 Problem # 36 Let A = 2 2 b = a. Use the Gram-Schmidt process to find an orthonormal basis for the column space of A. b. Factor A into QR where the columns of Q correspond the orthonormal basis in part (a). c. Solve Ax = b using the QR decomposition. Problem # 37 The vectors x = 2 [,,, ]T and x 2 = 6 [,, 3, 5]T form an orthonormal set in R 4. Extend this set to an orthonormal basis for R 4 by finding an orthonormal basis for the nullspace of [ ] 3 5 Hint: First find a basis for the nullspace and then use the Gram-Schmidt process. Problem # 38 Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R 4 spanned by x = [2, 0, 0, 2] T and x 2 = [,,, ] T. Problem # 39 Find the linear least squares solution for the data points (2, 3), (3, 2.5), (4, 2.25), (5, 2).

22

### MAT Linear Algebra Collection of sample exams

MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system

### Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination

Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column

### 2. Every linear system with the same number of equations as unknowns has a unique solution.

1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

### 1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det

What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix

### Problem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show

MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,

### Review problems for MA 54, Fall 2004.

Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on

### Linear 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.....................................

### MA 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

### Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial

### CSL361 Problem set 4: Basic linear algebra

CSL361 Problem set 4: Basic linear algebra February 21, 2017 [Note:] If the numerical matrix computations turn out to be tedious, you may use the function rref in Matlab. 1 Row-reduced echelon matrices

### MATH 240 Spring, Chapter 1: Linear Equations and Matrices

MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear

### 235 Final exam review questions

5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an

### MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left

### 1. In this problem, if the statement is always true, circle T; otherwise, circle F.

Math 1553, Extra Practice for Midterm 3 (sections 45-65) Solutions 1 In this problem, if the statement is always true, circle T; otherwise, circle F a) T F If A is a square matrix and the homogeneous equation

### Math Linear Algebra Final Exam Review Sheet

Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

### Solutions to Final Exam

Solutions to Final Exam. Let A be a 3 5 matrix. Let b be a nonzero 5-vector. Assume that the nullity of A is. (a) What is the rank of A? 3 (b) Are the rows of A linearly independent? (c) Are the columns

### Linear Algebra- Final Exam Review

Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.

### 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

### Conceptual Questions for Review

Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.

### Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called

### Math 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

### PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.

Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence

### MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018

Homework #1 Assigned: August 20, 2018 Review the following subjects involving systems of equations and matrices from Calculus II. Linear systems of equations Converting systems to matrix form Pivot entry

### (b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).

.(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)

### Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the

### Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.

Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read

### 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

### LINEAR ALGEBRA QUESTION BANK

LINEAR ALGEBRA QUESTION BANK () ( points total) Circle True or False: TRUE / FALSE: If A is any n n matrix, and I n is the n n identity matrix, then I n A = AI n = A. TRUE / FALSE: If A, B are n n matrices,

### 1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?

. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in

### Linear 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

### and 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) =

### MATH 304 Linear Algebra Lecture 34: Review for Test 2.

MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1

### Chapters 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

### 5.) 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

### Math 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

### Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures

### LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS

LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,

### Problem 1: Solving a linear equation

Math 38 Practice Final Exam ANSWERS Page Problem : Solving a linear equation Given matrix A = 2 2 3 7 4 and vector y = 5 8 9. (a) Solve Ax = y (if the equation is consistent) and write the general solution

### MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.

MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:

### 1. 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 +

### Chapter 5 Eigenvalues and Eigenvectors

Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n

### Math Computation Test 1 September 26 th, 2016 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge!

Math 5- Computation Test September 6 th, 6 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge! Name: Answer Key: Making Math Great Again Be sure to show your work!. (8 points) Consider the following

### LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for

### The eigenvalues are the roots of the characteristic polynomial, det(a λi). We can compute

A. [ 3. Let A = 5 5 ]. Find all (complex) eigenvalues and eigenvectors of The eigenvalues are the roots of the characteristic polynomial, det(a λi). We can compute 3 λ A λi =, 5 5 λ from which det(a λi)

### Lecture Notes in Linear Algebra

Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................

### MATH 2360 REVIEW PROBLEMS

MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1

### Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017

Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...

### MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION

MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether

### Study Guide for Linear Algebra Exam 2

Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real

### This MUST hold matrix multiplication satisfies the distributive property.

The columns of AB are combinations of the columns of A. The reason is that each column of AB equals A times the corresponding column of B. But that is a linear combination of the columns of A with coefficients

### Linear algebra II Tutorial solutions #1 A = x 1

Linear algebra II Tutorial solutions #. Find the eigenvalues and the eigenvectors of the matrix [ ] 5 2 A =. 4 3 Since tra = 8 and deta = 5 8 = 7, the characteristic polynomial is f(λ) = λ 2 (tra)λ+deta

### ANSWERS (5 points) Let A be a 2 2 matrix such that A =. Compute A. 2

MATH 7- Final Exam Sample Problems Spring 7 ANSWERS ) ) ). 5 points) Let A be a matrix such that A =. Compute A. ) A = A ) = ) = ). 5 points) State ) the definition of norm, ) the Cauchy-Schwartz inequality

### HOMEWORK 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

### MATH 1553, Intro to Linear Algebra FINAL EXAM STUDY GUIDE

MATH 553, Intro to Linear Algebra FINAL EXAM STUDY GUIDE In studying for the final exam, you should FIRST study all tests andquizzeswehave had this semester (solutions can be found on Canvas). Then go

### Extra Problems for Math 2050 Linear Algebra I

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

### 1. 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

### (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

### Online Exercises for Linear Algebra XM511

This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2

### Solutions 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

### SOLUTION KEY TO THE LINEAR ALGEBRA FINAL EXAM 1 2 ( 2) ( 1) c a = 1 0

SOLUTION KEY TO THE LINEAR ALGEBRA FINAL EXAM () We find a least squares solution to ( ) ( ) A x = y or 0 0 a b = c 4 0 0. 0 The normal equation is A T A x = A T y = y or 5 0 0 0 0 0 a b = 5 9. 0 0 4 7

### Math 21b. Review for Final Exam

Math 21b. Review for Final Exam Thomas W. Judson Spring 2003 General Information The exam is on Thursday, May 15 from 2:15 am to 5:15 pm in Jefferson 250. Please check with the registrar if you have a

### Final Examination 201-NYC-05 - Linear Algebra I December 8 th, and b = 4. Find the value(s) of a for which the equation Ax = b

Final Examination -NYC-5 - Linear Algebra I December 8 th 7. (4 points) Let A = has: (a) a unique solution. a a (b) infinitely many solutions. (c) no solution. and b = 4. Find the value(s) of a for which

### ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

### Chapter 3. Determinants and Eigenvalues

Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory

### Linear Algebra Highlights

Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to

### MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)

### A Brief Outline of Math 355

A Brief Outline of Math 355 Lecture 1 The geometry of linear equations; elimination with matrices A system of m linear equations with n unknowns can be thought of geometrically as m hyperplanes intersecting

### LINEAR ALGEBRA KNOWLEDGE SURVEY

LINEAR ALGEBRA KNOWLEDGE SURVEY Instructions: This is a Knowledge Survey. For this assignment, I am only interested in your level of confidence about your ability to do the tasks on the following pages.

### 2018 Fall 2210Q Section 013 Midterm Exam II Solution

08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique

### Eigenvalues and Eigenvectors A =

Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector

### (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)

### PRACTICE PROBLEMS FOR THE FINAL

PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show

### Review Let A, B, and C be matrices of the same size, and let r and s be scalars. Then

1 Sec 21 Matrix Operations Review Let A, B, and C be matrices of the same size, and let r and s be scalars Then (i) A + B = B + A (iv) r(a + B) = ra + rb (ii) (A + B) + C = A + (B + C) (v) (r + s)a = ra

### ELEMENTARY 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

### MATH 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

### ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST

me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following

### MTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education

MTH 3 Linear Algebra Study Guide Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education June 3, ii Contents Table of Contents iii Matrix Algebra. Real Life

### Dimension. Eigenvalue and eigenvector

Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,

### MATH 235. Final ANSWERS May 5, 2015

MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your

### MATH 15a: Linear Algebra Practice Exam 2

MATH 5a: Linear Algebra Practice Exam 2 Write all answers in your exam booklet. Remember that you must show all work and justify your answers for credit. No calculators are allowed. Good luck!. Compute

### Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.

Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the

### Linear Algebra. and

Instructions Please answer the six problems on your own paper. These are essay questions: you should write in complete sentences. 1. Are the two matrices 1 2 2 1 3 5 2 7 and 1 1 1 4 4 2 5 5 2 row equivalent?

### Math 310 Final Exam Solutions

Math 3 Final Exam Solutions. ( pts) Consider the system of equations Ax = b where: A, b (a) Compute deta. Is A singular or nonsingular? (b) Compute A, if possible. (c) Write the row reduced echelon form

### Third Midterm Exam Name: Practice Problems November 11, Find a basis for the subspace spanned by the following vectors.

Math 7 Treibergs Third Midterm Exam Name: Practice Problems November, Find a basis for the subspace spanned by the following vectors,,, We put the vectors in as columns Then row reduce and choose the pivot

### Math Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT

Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear

### TBP MATH33A Review Sheet. November 24, 2018

TBP MATH33A Review Sheet November 24, 2018 General Transformation Matrices: Function Scaling by k Orthogonal projection onto line L Implementation If we want to scale I 2 by k, we use the following: [

### 1 Last time: least-squares problems

MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that

### Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.

Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for

### Lecture Summaries for Linear Algebra M51A

These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture

### Linear algebra II Homework #1 solutions A = This means that every eigenvector with eigenvalue λ = 1 must have the form

Linear algebra II Homework # solutions. Find the eigenvalues and the eigenvectors of the matrix 4 6 A =. 5 Since tra = 9 and deta = = 8, the characteristic polynomial is f(λ) = λ (tra)λ+deta = λ 9λ+8 =

### Final Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2

Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch

### YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions

YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For

### 4. Determinants.

4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.

### [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.]

Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the

### Topic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C

Topic 1 Quiz 1 text A reduced row-echelon 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 correct-choice may have 0, 1, 2, or 3 choice may have 0,

### Chapter 3 Transformations

Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases

### MIT Final Exam Solutions, Spring 2017

MIT 8.6 Final Exam Solutions, Spring 7 Problem : For some real matrix A, the following vectors form a basis for its column space and null space: C(A) = span,, N(A) = span,,. (a) What is the size m n of

### MATH. 20F SAMPLE FINAL (WINTER 2010)

MATH. 20F SAMPLE FINAL (WINTER 2010) You have 3 hours for this exam. Please write legibly and show all working. No calculators are allowed. Write your name, ID number and your TA s name below. The total