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 and row Free variable Row operations, Gaussian elimination Row echelon form, Reduced row echelon form Vectors Matrix addition, subtraction, multiplication, scalar multiplication Symmetric matrix Matrix transpose Identity matrix Singular, nonsingular Matrix inverse, invertible matrices Determinants On homework or exams I expect you to tell me which row operations you are using. following notation can be used to state a row operation: The Interchange rows 1 and 3 R 1 R 3 Multiply row 2 by -7 7R 2 R 2 Replace row 3 by the sum of row 3 and 5 times row 4 5R 4 + R 3 R 3 This notation tells us which rows were used and which rows were ultimately changed. 1. Solve the system using row operations to put the associated matrix in strictly triangular form and then back substitute. 3x 1 + 2x 2 + x 3 = 0 2x 1 + x 2 x 3 = 2 2x 1 x 2 + 2x 3 = 1 1
2. Which of the following matrices are in row echelon form? Which are in reduced row echelon form? [ 1 2 3 ] 4 0 0 1 2 (b) 1 0 0 0 0 0 0 0 1 1 3 0 0 0 1 0 0 0 (d) 0 1 0 0 0 0 (e) 1 1 1 0 1 2 0 0 3 (f) 1 4 6 0 0 1 0 1 3 (g) 1 0 0 1 2 0 1 0 2 4 0 0 1 3 6 (h) 0 1 3 4 0 0 1 3 0 0 0 0 3. For each of the systems of equations that follow, use Gaussian elimination to obtain a solution involving reduced row echelon form. If there are infinite solutions, identify the free variables and describe all solutions. (b) x 1 2x 2 = 3 2x 1 x 2 = 9 2x 1 + 3x 2 + x 3 = 1 x 1 + x 2 + x 3 = 3 3x 1 + 4x 2 + 2x 3 = 4 x 1 2x 2 = 3 2x 1 + x 2 = 1 5x 1 + 8x 2 = 4 2
4. If compute the following quantities. 3 1 4 1 0 2 A = 2 0 1 and B = 3 1 1, 1 2 2 2 4 1 2A (b) A + B 2A 3B (d) (2A) T (3B) T (e) AB (f) BA (g) A T B T (h) (BA) T 5. For each pair of matrices, determine whether it is possible to perform the multiplication. If it is possible, compute the product. [ ] 1 3 5 1 2 1 3 2 0 2 4 1 (b) 4 2 6 4 [ 1 2 3 ] 8 6 1 4 3 3 2 0 1 4 1 1 0 0 2 4 5 (d) [ ] [ ] 4 6 3 1 5 2 1 4 1 6 (e) [ ] [ ] 4 6 1 3 1 5 2 1 1 4 1 6 (f) 2 1 [ 3 2 4 5 ] 3 6. Let A and B be symmetric n n matrices. For each of the following, determine whether the given matrix will be symmetric or non-symmetric. It will be helpful to know that the (i, j)-entry of a matrix M = AB, denoted m ij, can be computed with the sum n m ij = a ik b kj, k=1 where A is an m n matrix, B is an n r matrix, and M is an m r matrix. Another hint is that A T = A, or that a ij = a ji. C = A + B (b) D = A 2 E = AB (d) F = ABA (e) G = AB + BA (f) H = AB BA 3
7. Let C be a non-symmetric n n matrix. For each of the following, determine whether the given matrix must be symmetric or could be non-symmetric. A = C + C T (b) B = C C T D = C T C (d) E = C T C CC T (e) F = (I + C)(I + C T ) (f) G = (I + C)(I C T ) 8. Evaluate the following determinants by hand. 3 5 2 3 (b) 5 2 8 4 3 1 2 2 4 5 2 4 5 (d) 4 3 0 3 1 2 5 1 4 (e) 1 3 2 4 1 2 2 1 3 (f) 2 1 2 1 3 2 5 1 6 9. Let A and B be 2 2 matrices. Justify your answers by either proving the statement is true or by providing a counterexample. Does det (A + B) = det (A) + det (B)? (b) Does det (AB) = det (A) det (B)? Does det (AB)) = det (BA)? (d) Does det (A T ) = det (A)? 4
Homework #2 Assigned: August 24, 2018 1. Let V = R, so the vectors are real numbers. To remind us that vectors are real numbers in this example, feel free to not use vector notation (use x instead of x). Define scalar multiplication as usual αx = α x, and define addition, denoted by, by x y = max {x, y}. Is V a vector space? Clearly spell out why or why not (every property needs explanation). Assume nothing is clear to the reader and everything needs to be explained. Also, use the names of the properties you are checking along with the numbers we use in class for the sake of clarity. Since you need the practice, make sure you address all ten properties. 2. Determine whether the following sets form a subspace of R 3. {[x 1, x 2, x 3 ] T : x 1 + x 3 = 1} (b) {[x 1, x 2, x 3 ] T : x 1 = x 2 = x 3 } 3. Determine whether the following sets form a subspace of R 2 2. The set of all 2 2 lower triangular matrices. You may need to find out what this means. Try www.mathworld.com. (b) The set of all 2 2 matrices A such that a 11 = 1. The set of all 2 2 matrices B such that b 11 = 0. (d) The set of all symmetric 2 2 matrices. (e) The set of all singular 2 2 matrices. The following theorem will be helpful. Theorem 1. An n n matrix A is singular if and only if det (A) = 0. 4. Determine whether the following sets form a subspace of P 4. The set of all polynomials in P 4 of even degree. (b) The set of all polynomials p(x) P 4 such that p(0) = 0. 5. Determine whether the following sets form a subspace of C[ 1, 1]. The set of odd functions in C[ 1, 1]. Recall that a function is odd if f( x) = f(x) for all x in its domain, which is [ 1, 1]. (b) The set of functions in C[ 1, 1] such that f( 1) = 0 and f(1) = 0. The set of functions in C[ 1, 1] such that f( 1) = 0 or f(1) = 0. 5
6. Let V be a vector space and let x, y, z V. Prove the following: If x + y = x + z, then y = z. (b) β0 = 0 for every β R. Use the fact that β0 = β(0 + 0). If αx = 0 and α is nonzero, then x = 0. Use (b). 6
Homework #3 Assigned: September 7, 2018 1. Let A R n n, and define the set C(A) to be the set of all matrices that commute with A, or Show C(A) is a subspace of R n n. C(A) = {B : B R n n and AB = BA}. 2. In each of the following, determine the subspace of R 2 2 consisting of all matrices that commute with the given matrix. As an example, which you might want to work out for yourself so that you understand the process before trying the other problems, consider ([ ]) {[ ] } 1 0 b11 0 C = : b 0 1 0 b 11, b 22 R. 22 [ ] 0 0 1 0 [ ] 1 1 (b) J = 0 1 [ ] 1 1 1 1 3. Let A R 2 2. Determine whether the following sets are subspaces of R 2 2. S 1 = {B R 2 2 : BA = O}, where O is the matrix of all zeros. (b) S 2 = {B R 2 2 : AB BA} S 3 = {B R 2 2 : AB + B = O} 4. Given 1 3 x 1 = 2 x 2 = 4 3 2 2 9 u = 6 v = 2, 6 5 answer the following questions. Justify your answers by writing out the necessary linear combination or explaining why no linear combination is possible. Is u Span (x 1, x 2 )? (b) Is v Span (x 1, x 2 )? 7
5. Determine whether the following are spanning sets for R 2. Justify your answers. {[ ] [ ]} 2 4, 3 6 (b) {[ ] 2, 1 [ ] 1, 3 [ ]} 2 4 6. Determine whether the following sets are spanning sets for P 3. Justify your answers. {1, x 2, x 2 2} (b) {x + 2, x + 1, x 2 1} 8
Homework #4 Assigned: September 17, 2018 If you find the need to row reduce a matrix, feel free to use technology to do so. If you do use technology, report your work like I do in class: state the matrix to be reduced and its final reduced state, but do not feel the need to provide the intermediate steps. 1. Determine whether the following vectors are linearly independent in R 3. (b) 1 0 1 0, 1, 0 0 1 1 2 2 4 1, 1, 2 2 2 4 1 0 1, 2 3 1 2. Determine whether the following vectors are linearly independent in P 3. x + 2, x + 1, x 2 1 (b) x + 2, x 2 1 3. Let x 1, x 2, and x 3 be linearly independent vectors in R n, and let y 1 = x 2 x 1, y 2 = x 3 x 2, y 3 = x 3 x 1. Are y 1, y 2, and y 3 linearly independent? Justify your answer. 4. Let A be an m n matrix. Prove that if A has linearly independent column vectors, then N(A) = {0}. Hint: for any x R n, Ax = x 1 a 1 + x 2 a 2 + + x n a n, where a i is the i-th column of A. 5. Let S and T be subspaces of a vector space V. Prove that their intersection, S T = {v : v S and v T }, is a subspace of V. 9
Homework #5 Assigned: September 26, 2018 If you find the need to row reduce a matrix, feel free to use technology to do so. If you do use technology, report your work like I do in class: state the matrix to be reduced and its final reduced state, but do not feel the need to provide the intermediate steps. 1. Find a basis for the subspace S of R 4 consisting of all vectors of the form a + b a b + 2c b, c where a, b, c R. What is the dimension of S? Justify your answers. 2. The vectors 1 2 1 2 1 x 1 = 2, x 2 = 5, x 3 = 3, x 4 = 7, x 5 = 1 2 4 2 4 0 span R 3. Pare down the set {x 1, x 2, x 3, x 4, x 5 } to form a basis of R 3. Justify your answer. 3. Let S be the subspace of P 3 consisting of all polynomials p(x) such that p(0) = 0, and let T be the subspace of all polynomials q(x) such that q(1) = 0. Find bases for the following sets. S (b) T S T 4. Recall an early example of a vector space: {[ ] } x1 V = : x 1, x 2 R, x 2 > 0, x 2 where for α R and x, y V, [ ] αx1 α x = x α 2 [ ] x1 + y and x y = 1. x 2 y 2 Do the vectors [ ] [ ] 1 1, 2 4 form a basis for V? 10
Homework #6 Assigned: October 19, 2018 1. Consider the following bases for R 2 : E = {e 1, e 2 }, U = {u 1, u 2 }, and W = {w 1, w 2 }, where u 1 = [ ] 1, u 2 2 = Find the transition matrix from U to E. (b) Find the transition matrix from E to U. [ ] 3 Convert x = to [x] 2 U. [ ] 2, w 5 1 = (d) Find the transition matrix from W to U. [ ] 1 (e) Convert [x] W = to [x] 4 U. 2. Given W v 1 = [ ] 2, v 6 2 = [ ] 3, w 2 2 = [ ] 1, and S 4 VU = [ ] 4. 3 [ ] 4 1 2 1 find vectors u 1 and u 2 such that S is the transition matrix from V = {v 1, v 2 } to U = {u 1, u 2 }. 3. Let E = {1, x} and F = {2x 1, 2x + 1} be bases for P 2. Find the transition matrix from F to E. (b) Find the transition matrix from E to F. [ ] 3 Convert x = to [x] 1 E. Check that your answer is correct. F 4. Several collections of vectors exist that are used as bases for various vector spaces involving functions. Two such collections are the Laguerre polynomials, L, and the Hermite polynomials, H. Below are the first four polynomials for each collection, which each form a basis for P 4. L = {1, 1 t, 2 4t + t 2, 6 18t + 9t 2 t 3 } H = {1, 2t, 2 + 4t 2, 12t + 8t 3 } Prove that both of these collections form a basis for P 4. (b) Find the transition matrix from L to H. Convert 3 + t 6t 2 to both L and H. Note that the given polynomial is written in the standard basis for P 4. 11
Homework #7 Assigned: October 29, 2018 1. Find a basis for the row space, column space, and null space for A. Provide dimensions for each space and verify that the dimensions of the row space and column space are equal. Also, verify the Rank-Nullity Theorem. 3 1 3 4 A = 1 2 1 2 3 8 4 2 2. Let A be a 6 n matrix of rank r, and let b be a vector in R 6. For each pair of values r and n that follow, indicate the possibilities as to the number of solutions one could have for the linear system Ax = b. Explain your answers. n = 7, r = 5 (b) n = 7, r = 6 n = 5, r = 5 (d) n = 5, r = 4 3. Determine whether the following are linear transformations. [ ] 1 + L 1 : R 3 R 2 x1, where L 1 (x) =. [ ] (b) L 2 : R 3 R 2 x, where L 2 (x) = 3. x 1 + x 2 x 2 L 3 : R n n R n n, where L 3 (A) = A T. (d) L 4 : R n n R n n, where L 4 (A) = A A T. 4. Let C be a fixed n n matrix. Determine whether the following are linear transformations from R n n to R n n. L 5 (A) = CA + AC (b) L 6 (A) = C 2 A L 7 (A) = A 2 C 5. Find the kernel and range of each of the following linear transformations from P 3 to P 3. L 8 (p(x)) = xp (x). (b) L 9 (p(x)) = p(x) p (x) L 10 (p(x)) = p(0)x + p(1). 6. Determine the kernel and range of each of the following linear transformations on R 3. L 11 (x) = [x 3, x 2, x 1 ] T 12
(b) L 12 (x) = [x 1, x 1, x 1 ] T 7. Let S be the subspace of R 3 spanned by e 1 and e 2. For each linear transformation in Problem 6, find L(S). 8. Let L : V W be a linear transformation, and let T be a subspace of W. The inverse image of T, denoted L 1 (T ), is defined by L 1 (T ) = {v V : L(v) T }. Show that L 1 (T ) is a subspace of V. (Note that L 1 (T ) is a set and that there is no inverse linear transformation L 1 : W V.) 9. For each of the following linear transformations L : R 3 R 2, find a matrix A such that L(x) = Ax for every x R 3. L 13 (x) = [x 1 + x 2, 0] T (b) L 14 (x) = [x 1, x 2 ] T L 15 (x) = [x 2 x 1, x 3 x 2 ] T 10. Find the standard matrix representation for each of the following linear transformations. L 16 is the linear transformation that rotates each x in R 2 by 45 in the clockwise direction. (b) L 17 is the linear transformation that reflects each vector x in R 2 about the x 1 axis and then rotates it 90 in the counterclockwise direction. L 18 is the linear transformation that doubles the length of x in R 2 and then rotates it 30 in the counterclockwise direction. (d) L 19 is the linear transformation that reflects each vector x in R 2 about the line x 2 = x 1 and then projects it onto the x 1 -axis. 13
Homework #8 Assigned: November 9, 2018 1. Find the eigenvalues and the corresponding eigenspaces for each of the following matrices. Show your work, though you may use technology to perform row reductions. Also state the algebraic and geometric multiplicities for each eigenvalue. (b) (d) A = B = [ ] 3 1 1 1 [ ] 3 8 2 3 2 0 1 C = 1 0 1 0 1 1 1 1 1 D = 0 2 1 0 0 1 2. Let A be an n n matrix, and let B = I 2A + A 2. Show that if x is an eigenvector of A belonging to an eigenvalue λ of A, then x is also an eigenvector of B belonging to an eigenvalue µ of B. How are λ and µ related? (b) Show that if λ = 1 is an eigenvalue of A, then the matrix B will be singular. 3. An n n matrix A is an idempotent if A 2 = A. Show that if λ is an eigenvalue of an idempotent matrix, then λ must be either 0 or 1. 4. Let A be a 2 2 matrix. If tr (A) = 8 and det A = 12, what are the eigenvalues of A? 5. Let A be an n n matrix, and let λ be an eigenvalue of A. Prove that E λ (A) is a subspace of R n. 6. Let A be an n n matrix and λ an eigenvalue of A. If A λi has rank k, what is the geometric multiplicity of E λ (A)? Explain your answer. 7. Diagonalize the following matrices, if possible. A = [ ] 2 8 1 4 14
(b) 1 0 0 B = 2 1 3 1 1 1 8. The city of Mawtookit maintains a constant population of 300,000 people from year to year. A political science study estimated that there were 150,000 Independents, 90,000 Democrats, and 60,000 Republicans in the town. It was also estimated that each year 20% of the Independents become Democrats and 10% become Republicans. Similarly, 20% of Democrats become Independents and 10% become Republicans. Finally, 10% of Republicans become Democrats and 10% become Independents. Find the transition matrix, M, for the Markov chain outlined above. (b) Diagonalize the matrix M found in. Which group will dominate town politics in the long run? Justify your answer. 15