MATH 223 Assignment #1 1. Let A = 0 x 1 y and B = 3 1 1 1 . Determine all x, y so that AB = BA. 2. Find a 2 ⇥ 2 matrix A, no entry of which is 0, with A2 = A. Note that your first guesses A = I or A = 0 (or indeed A = E11) have 0 entries. 3. Darryl has a sum of s dollars in t coins. The coins are either $.25 or $.10 Express the number x of quarters ($.25 coins) and the number y of dimes ($.1 coins) in terms of s and t. 4. Assume you are given a pair of matrices A,B which satisfy AB = BA. Show that if we set C = A2 + 2A and D = B3 + 5I, then CD = DC. Then try to generalize this in some interesting way, namely find a property so that for matrices C,D with that certain property, then CD = DC. For example C = A2 + 6A and D = 3B3 2I will also have CD = DC. 5. Let R(✓) denote the matrix of the transformation which rotates the plane by ✓ counterclockwise around the origin. Explain in words (using transformations) why R(✓)R( ) = R(✓ + ). Show how you can use this to derive the formulas for cos(✓ + ), sin(✓ + ) in terms of cos(✓), sin(✓), cos( ), sin( ). 6. Find the matrix A associated with the linear transformation T that has T 1 1 = 2 3 and T 2 3 = 5 4 . 7. a) Assume A,B are 2⇥ 2 invertible matrices so that A 1 and B 1 exist. Show that (AB) 1 = B 1A 1. b) Given A = a b c d then define AT = a c b d , where AT is called the transpose of A. The dot product of two vectors x = a b , y = c d is x · y = ac + bd. Then the i, j entry of AB is the dot product of the ith row of A and the jth column of B. Using this idea, show that (AB)T = BTAT . (One could verify (AB)T = BTAT for two arbitrary 2⇥ 2 matrices A,B directly but the argument wouldn’t generalize to larger matrices). 8. Consider two nonzero vectors x = a b , y = c d . Then there is a ✓ with 0 ✓ < 2⇡ and a ⇢ > 0 so that y = ⇢R(✓)x. Use our knowledge of rotation matrices to establish a simple condition on a, b, c, d so that the angle ✓ satisfies 0 < ✓ < ⇡. You may assume a, b, c, d are nonzero, if that assists you, and even assume the two vectors x,y have the same length (⇢ = 1). MATH 223 Assignment #2 1. Let A1 = " 5 6 1 0 # and A2 = " 1 1 1 3 # . For each of these two matrices, determine the eigen- values and for each eigenvalue determine an eigenvector. For A2 the eigenvalues are a little more complicated making the computations a little harder. Then give the diagonalization of each matrix; namely an invertible matrixM and a diagonal matrix D with AM = MD. (The equation AM = MD is important because it will yield A = MDM 1 and M 1AM = D). 2. Let A be a 2 ⇥ 2 matrix with two di↵erent eigenvalues 1, 2 and associated eigenvectors v1,v2. Let v = av1 + bv2. Assume that | 1| > | 2|. Show that lim n!1 Anv n1 = av1. How do you define the limit? For a 6= 0 this means that we see the eigenevector v1 appearing in the limit. 3. Review the notes on Fibonacci numbers. Let f1, f2 be two arbitrary integers, not both zero. Consider the sequence f1, f2, f3, f4, . . . where fi = fi 1 + fi 2 for i = 3, 4, 5, . . .. We wish to show that lim n!1 fn fn 1 = 1 + p 5 2 . Firstly, explain why we can solve for c1, c2 in the vector equation" f2 f1 # = c1 " 1+ p 5 2 1 # + c2 " 1 p5 2 1 # . Using our hypothesis that f1, f2 are not both zero, we deduce that c1, c2 are not both zero. Secondly, use our hypothesis that f1, f2 are integers, not both zero, to deduce c1 6= 0. The irrationality of p 5 (which you need not prove) combined with c1, c2 being integers is impor- tant. Thirdly verify the limit. If you can’t show c1 6= 0 then you can still proceed assuming c1 6= 0 to establish this limit. Hint: use ideas of the previous question. 4. In this question, we explore the behaviour of An when A does not have distinct eigenvectors (up to rescaling). (a) Let A := " 1 1 0 1 # . i. Find all eigenvectors and eigenvalues of A. ii. Give a simple expression for An. (b) Consider a set of 2-tuples satisfying" xn+1 yn+1 # = " 3 1 1 1 # " xn yn # , n = 0, 1, 2, ... Let x0 = y0 = 1. Give a relatively simple expression for xn + yn. MATH 223 Assignment #3 1. I wish to see the solutions to a system of equations in Parametric Vector Form (or Vector Parametric Form). For eample if the set of solutions is: x1 = 3r 4s 2t x2 = r x3 = 2s x4 = s x5 = t x6 = 1/3 for all choices r, s, t 2 R then we can write the set of solutions in parametric vector form as follows: 2666666664 x1 x2 x3 x4 x5 x6 3777777775 = 2666666664 0 0 0 0 0 1/3 3777777775 + r 2666666664 3 1 0 0 0 0 3777777775 + s 2666666664 4 0 2 1 0 0 3777777775 + t 2666666664 2 0 0 0 1 0 3777777775 , r, s, t 2 R. Give the vector parametric form of all solutions to the following system of equations: 2x1 +4x4 +6x5 = 14 2x1 +5x4 +7x5 = 16 3x1 +2x2 +8x4 +9x5 = 27 3x1 +4x2 +13x4 +12x5 = 39 2. Give the solutions in vector parametric form for the plane ⇡ = {(x, y, z) : 2x 2y+3z = 5}. 3. Express the inverse of the following matrix A as a product of elementary matrices. A = 264 1 1 02 3 1 0 0 1 375 4. Compute i) det 264 x 0 010 x 0 52 223 1 375 , ii) det 264 99 100 1010 0 0 4 e 98 375 , iii) det 26664 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 37775 MATH 223 Assignment #4 (two pages) 1. On future quizzes, you may be expected to compute the eigenvalues and associated eigenvec- tors for a 3 ⇥ 3 matrix. We did the 2 ⇥ 2 case in class. You’ll need to figure out how this generalizes. Let A = 24 1 2 12 2 2 1 2 1 35 a) Find the eigenvectors of eigenvalue 2. (This should be just Gaussian Elimination) b) Compute det(A I) (a cubic polynomial in ) by the expansion method. The leading term in the cubic polynomial is 3. I’d recommend pulling out a factor of -1. A check on your work is that ( 2) should be a factor of the polynomial (why? because 2 is a root). c) Factorize det(A I) and determine all eigenvalues and for each eigenvalue, describe the associated set of eigenvectors. 2. Assume A is a 3 ⇥ 3 matrix, and M is an invertible matrix with A = MDM 1, where D is the diagonal matrix D = 24 2 0 00 3 0 0 0 4 35 Show that (A 2I)(A 3I)(A 4I) = 0 where 0 denotes the 3⇥ 3 matrix of 0’s. 3. Let A = (aij) be an n⇥ n matrix with integral entries such that the diagonal entries are all not divisible by 3 (aii is not evenly divisible by 3) and all o↵ diagonal entries are divisible by 3 (aij is divisible by 3 for i 6= j). Show that A has det(A) 6= 0, i.e. A is invertible. 4. Are there functions, f : Rn⇥n ! R, satisfying the following two requirements instead? If so, describe them. (a) f(AB) = f(A) + f(B). (b) f(A) 6= 0 if and only if A is invertible. 5. Let A be an n by n matrix and suppose that A has n distinct eigenvalues. For simplicity, assume they are ordered and positive real numbers so that 1 > 2 > 3 > ... > n 0. Suppose the eigenvalues are associated to the eigenvectors v1, v2, ..., vn (in matching order). Let M be the matrix whose i-th column is vi. Note we have AM = MD where D is the diagonal matrix with the eigenvalues on the diagonal. (a) Must M be invertible? If so, prove it. If not, give an example where it is not. Hint: It may be helpful to use the fact that Mx = x1v1 + x2v2 + ...+ xnvn, where x = [x1, x2, ..., xn] T . (b) If M is invertible, note that A can be diagonalized as A = MDM 1. In that case, what is the relationship between det(A) and the eigenvalues of A? 6. (Challenge, not for credit. If you solve this, email the solution to the professor, the TA may like a copy as well.) We know that the determinant is a function f : Rn⇥n ! R, satisfying: (a) f(AB) = f(A) · f(B). (b) f(A) 6= 0 if and only if A is invertible. (c) f is continuous. (You may have to look up, or define yourself, what continuous means for functions of matrices.) Can you describe the set of all functions satisfying these requirements? If so, prove that you have described all such functions. (This hasn’t been solved by a student before.) MATH 223 Assignment #5 Note: For problems 1,2,3, let R be the field associated with each vector space. 1. For each of the following sets, circle T if it is a vector space (including the case when it is a subspace), and F if it is not. You do not need to show work for this problem. (The definition of addition and scalar multiplication for these sets follow the standard choices.) (a) {(b1, b2, b3) such that b1 = 1, b2, b3 2 R} T F (b) {(b1, b2, b3) such that 2b1 5b2 + b3 = 0, b1, b2, b3 2 R} T F (c) {(b1, b2, b3) such that b2b3 = 0, b1 2 R} T F (d) {(0, 0, 0)} T F (e) Infinite sequences {xi, i 1, such that xi+1 xi}. T F (f) The set of n by n matrices A which satisfy AT = A. T F (g) The set of n by n invertible matrices. T F (h) The set of 4 by 4 matrices with all eigenvalues greater than or equal to 0. T F (i) The set of polynomials with degree at least 3. T F 2. Which of the following are subspaces of the vector space of all functions f with domain R and range contained in R a) all f such that f( 1) = 0. b) all f such that f(x) 0 for all x 2 R. c) all f of the form f(x) = k1 + k2 sin(x) where k1, k2 2 R. 3. Consider the two dimensional vector space V = span(cos2(x), sin2(x)), a subspace of all functions from R ! R. Which of the following belong to V (the argument to show f /2 V will be more di cult). (a) 0 (b) 2 (c) 3 + x2 (d) cos(2x) 4. Show that 1 and p 2 are linearly independent when we restrict ourselves to the scalar field Q, the rational numbers. In other words show that there do not exist 4 integers a, b, c, d with b 6= 0, d 6= 0 and not both a = 0 and c = 0, which satisfy a b ⇥ 1 + c d ⇥p2 = 0. (You may need to prove that p 2 is irrational, don’t take that as given.) 5. This is a putnam problem. Let A be a 2013 ⇥ 2014 matrix of integer entries such that each row sum is 0 (i.e. A1 = 0 where 1 is the 2014⇥ 1 vector of 1’s and 0 is the 2013⇥ 1 vector of 0’s. Show that det(AAT ) = 2014k2 for some integer k. Hint: You might find it helpful to form a new square matrix B from A by adding a row of 1’s. What is det(BBT )? 6. Let V be a vector space over a field F . Then, given ↵ 2 F and v 2 V such that ↵v = 0, prove that either ↵ = 0 or v = 0. For each step, state which axiom you use. (This would be less necessary after we had more experience with abstract vector spaces.) MATH 223 Assignment #6 1. Let A 2 Rm⇥n have rank 1. Show that there exist non-zero vectors x 2 Rm and y 2 Rn so that A = xyT . (Hint: Try a simple case and also compute xyT for some simple choices for x and y.) (Comment: You could explore how to generalize such a result to higher rank.) 2. Determine bases for the following subspaces of R3. a) the line x = 5t, y = 2t, z = t. b) all vectors of the form (a, b, c)T such that a 3b = 2c. 3. Let A = 26664 0 1 1 2 3 1 0 2 0 6 6 0 0 3 7 2 9 7 0 2 2 4 4 3 37775 Determine a basis for the column space of A (chosen from columns of A) and determine a basis for the row space of A. Also give a basis for the nullspace of A, namely {x 2 R6 : Ax = 0}. 4. Show that the set of all vectors (b1, b2, b3, b4)T such that the system below is consistent (i.e. can be solved) 26664 2 3 1 4 3 3 1 3 0 2 0 2 37775x = 26664 b1 b2 b3 b4 37775 is a subspace of R4. Then find a basis of the subspace. 5. Let A be an n ⇥ n matrix with various eigenvalues including and µ with 6= µ. Let L,M be the eigenspaces associated with eigenvalues , µ respectively. (That is, L is the set of all eigenvectors with eigenvalue ; M is the set of all eigenvectors with eigenvalue µ.) Let {u1,u2, . . . ,up} be a basis for L and let {v1,v2, . . . ,vq} be a basis for M . Show that {u1,u2, . . . ,up,v1,v2, . . . ,vq} is a linearly independent set of p+ q vectors. (Hint: try p = 1 and q = 1 to start). (Comment: You could explore the case if there were three di↵erent eigenvalues and three bases for the eigenspaces). 6. Let Rn⇥n denote the vector space of all n ⇥ n matrices (over R). Consider following trans- formation f : Rn⇥n ! Rn⇥n f(A) = AT . Show that this is a linear transformation. We say that a matrix A is symmetric if AT = A and we say that a matrix A is skew-symmetric if AT = A. a) Warmup question: Give a basis for Rn⇥n. How many elements are in your basis? b) What is the dimension of the eigenspace of eigenvalue 1 for f? Explain. c) What is the dimension of the eigenspace of eigenvalue -1 for f? Explain. d) Now use the previous question (and other facts) to show that any A 2 Rn⇥n is a linear combination of a symmetric matrix and a skew-symmetric matrix (you could show this directly of course but I’m asking you to use linear independence/dimension arguments). MATH 223 Assignment #7 1. Let {u1,u2,u3} be a basis for a vector space V . Then if we define v1 = u1 + 2u3, v2 = u1 + 2u2 + 3u3, v3 = u2 u3, show that {v1,v2,v3} forms a basis for V . 2. (from a test) Let w1 = 264 12 0 375 , w2 = 264 25 1 375 , w3 = 264 24 1 375 NOTE: 264 1 2 22 5 4 0 1 1 375 1 = 264 1 0 2 2 1 0 2 1 1 375 Let f : R3 ! R3 be the linear transformation satisfying f(w1) = w2 w3, f(w2) = w2 +w3, f(w3) = w1 +w2 +w3. a) Give the matrix representation of f with respect to the basis {w1,w2,w3}. b) Give the matrix representation of f where the input x, is written with respect to the basis {w1,w2,w3} and the output f(x) is written with respect to the basis {e1, e2, e3} (the standard basis). c) Is w1 in the range of f? 3. (from a test) Let z1 = 264 20 1 375 , z2 = 264 11 0 375 , z3 = 264 0 1 1 375 NOTE: 264 2 1 00 1 1 1 0 1 375 1 = 264 1 1 11 2 2 1 1 2 375 Let T : R3 ! R3 be the linear transformation satisfying T (z1) = 2z2, T (z2) = 2z2, T (z3) = z1 + z2. a) Give the matrix representation of T with respect to the basis {z1, z2, z3}. b) Give the matrix representation of T with respect to the basis {e1, e2, e3} (the standard basis). Give the explicit matrix with integer entries. c) Give the matrix representing T 2 with respect to the basis {z1, z2, z3}. What is the rank of the matrix representing T 2 with respect to the standard basis {e1, e2, e3}? 4. Solve the system of di↵erential equations d dtx1(t) = 5x1(t) + 6x2(t) d dtx2(t) = 1x1(t) First find the general solution for x1(t), x2(t) as a function of x1(0), x2(0). Given x1(0) = 6 and x2(0) = 2, find the solutions explicitly and compute lim t!1 x1(t) x2(t) . Math 223 Assignment 8 1. Sove the di↵erential equation given the initial conditions x1(0) = 3, x2(0) = 4. d dtx1(t) = + x2(t) d dtx2(t) = 2x1(t) 2x2(t) 2. You are given a 3 dimensional vector space V ✓ R5. Could there be a 3 ⇥ 6 (not a typo) matrix A with nullspace of A being V ? Explain. Could there be 6⇥5 matrix B with nullspace of B being V ? Explain. In either case, if you were given a basis for the three dimensional space V , how would you find the desired matrix assuming it exists. 3. Consider the two planes ⇡1: x y + 2z = 3 and ⇡2: x+ 2y + 3z = 6. a) Find the intersection of ⇡1 and ⇡2 in vector parametric form. b) What is the angle (or just the cosine of the angle) formed by the two planes? (This is defined as the angle between their normal vectors. A normal vector is a vector orthogonal u v for every u, v in the plane.) c) Find the distance of the point ( 1, 2, 2) to the plane ⇡1. (That is, find the distance between ( 1, 2, 2) and the closest point in ⇡1.) d) Find the equation of the plane parallel to ⇡1 through the point (3, 2, 0). e) Imagine the direction (0, 0, 1)T as pointing straight up from your current position (0, 0, 0)T in 3-space and the plane ⇡2 as a physical plane. If a marble is placed on ⇡2 at the point (6, 0, 0)T , what direction will the marble roll under the influence of gravity? 4. Given a matrix A 2 Rn⇥n, we define the trace tr(A) = nX i=1 Ai,i, i.e., the sum of the diagonal. This is an important quantity. a) Let A,B 2 Rn⇥n. Show that tr(AB) = tr(BA). Hint: You may wish to express AB using the dot products between rows of A and columns of B. To be precise, let u1, u2, ..., un be the columns of AT (rows of A) and v1, v2, ..., vn be the columns of B. Then (AB)i,j = ui · vj. You can then show that tr(AB) is the dot product between AT and B (it’s up to you to define this dot product between matrices). b) Suppose that A can be diagonalized as A = MDM 1 where D is a diagonal matrix of eigenvalues 1, 2, ..., n. Show that tr(A) = nX i=1 i. Important note: The above equality is true even if A cannot be diagonalized. In other words, let 1, 2, ..., n be the n solutions to the characteristic equation det(A I) = 0. By the Fundamental Theorem of Algebra, there are always n solutions when counted with multiplicity. These are the eigenvalues of A. Then tr(A) = nX i=1 i. You may use this fact without proof. Math 223 Assignment 9 1. Find an orthonormal basis for R3 by applying Gram-Schmidt to the three vectors:264 12 2 375 , 264 43 2 375 , 264 12 1 375 . Please recall that we can change a vector with fractional entries to one with integral entries by rescaling and we do not need to normalize until the last step. 2. Please note the very important fact that an n ⇥ n symmetric matrix always is diagonalizable with an orthonormal basis of eigenvectors for Rn. This will be covered in class but perhaps not completed before the due date of this assignment. Find orthonormal bases for of eigenvectors for the following matrices : A = 1 2 2 4 , B = 264 0 10 1010 5 0 10 0 5 375 , C = 264 1 2 1 2 2 2 1 2 1 375 Hint: for B, 0 is an eigenvalue, and for C, 2 is an eigenvalue. 3. Let A be a 3⇥3 matrix with det(A I) = ( 3+a 2+b +c). Show that if A is diagonalizable, then the following equation is true A3 + aA2 + bA+ cI = 0 (this equation is in fact true for any 3 ⇥ 3 matrix and is a special case of the Cayley-Hamilton Theorem). 4. Let {u1,u2, . . . ,uk} be non-zero vectors satsifying ui · uj = 0 for all pairs i, j with i 6= j. Show that {u1,u2, . . . ,uk} are linearly independent. 5. Let A be an n⇥ n symmetric matrix with eigenvalues 1, 2, . . . , n (some may repeat) and an orthonormal basis of eigenvectors u1,u2, . . . ,un (Aui = iui). Then show that A = nX i=1 iuiu T i . (thus A is a sum of n symmetric rank 1 matrices) 6. (from an exam) Consider a matrix a b c d . We could attempt to solve for A 1 by letting A 1 = x y z t with four variables x, y, z, t and then AA 1 = I becomes a system of 4 equations in 4 unknowns with an associated 4 ⇥ 4 matrix B. What is the rank of the 4 ⇥ 4 system of equations assuming A 1 exists? Explain. Can you say anything about the rank of B if det(A) = 0? Explain. 5. Let A be a n⇥ n matrix of real entries satisfying A2 = I. Show that a) A is invertible (or nonsingular) b) A has no real eigenvalues c) n is even d) (harder question) det(A) = 1. (Hint: Try using the previous question.) 6. Consider two vectors spaces U, V , subspaces of Rm. Define U +V = {u+v : u 2 U,v 2 V }. (This is called the Minkowski sum.) Show that U + V is a vector space. Now show that dim(U) + dim(V ) = dim(U \ V ) + dim(U + V ). (Hint: if we have an m⇥ n matrix A then n = dim(nullspace(A)) + rank(A). How should we form A? ) Math 223 Assignment 10 The problems after Problem 1 play with singular value decomposition (svd). 1. You are attempting to solve for x, y, z in the matrix equation Ax = b where A = 2664 1 1 1 1 1 1 1 1 1 1 1 1 3775 , x = 24xy z 35 , b = 2664 3 2 1 0 3775 . Find a ‘least squares’ choice bˆ in the column space of A (and hence with ||b bˆ||2 being minimized) and then solve the new system Ax = bˆ for x, y, z. You can check your choice of bˆ by testing if b bˆ is orthogonal to the column space of A. 2. Let Q,W 2 R5⇥5 be orthogonal matrices. Let q1, q2, q3, q4, q5 be the columns of Q and w1, w2, w3, w4, w5 be the columns of of W . To be precise qi is the ith column of Q and wi is the ith column of W . Give a basis for the null space of the following matrices, which are almost in SVD form: (a) (5 pts) A = Q⌃W T , ⌃ = 266664 10 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 377775 (b) (5 pts) B = QSW T , S = 266664 0 0 0 3 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 377775 3. Let A be a matrix with SVD A = U⌃V T (this is the standard SVD, not the reduced SVD). Suppose that ⌃ = 241 0 0 00 2 0 0 0 0 3 0 35 . A right-inverse of A is a matrix B which satisfies AB = I. Does A have two or more right inverses? If so, find two right inverses. If not, why not? Hint: You may write your answer to the above question in terms of U and V . For example, if the question had been, what is ATA? Then the answer would have been ATA = V ⌃T⌃V T = V 2664 1 0 0 0 0 4 0 0 0 0 9 0 0 0 0 0 3775V T . 4. (10 pts) (This question is inspired by the idea of noise shaping – the fact that in some applications one can force the noise to take a certain pattern. ) Suppose that you have data from the (noiseless) linear model, but it becomes corrupted in the following way: One fixed, but unknown, constant is added to each entry of the data. To be precise, assume the following model: y = Ax+ z, z = 2664 c c c c 3775 . The matrix A has the singular value decomposition A = U⌃V T , U = 2664 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 3775 , ⌃ = 2664 3 0 0 0 2 0 0 0 1 0 0 0 3775 , V 2 R3⇥3 is orthogonal. You observe y and A, but not c or x. Is there a linear way to determine x? To be precise, is there a matrix W satisfying Wy = x no matter what c is? If so, determine the matrix W (the answer could be written in terms of V ). 5. Let U and V be two 5-dimensional subspaces of R9. Show that there is a nonzero vector in U \ V , the intersection of U and V . You may find it helpful to consider the case of two 2-dimensional subspaces of R3 first but you won’t be able to use ideas of lines and planes in R9. 6. The following question (Part c) is from a Putnam exam. This version has some added hints (Parts a and b) to perhaps make it doable. a) Show that it is impossible to have vectors u,v,w, t with26664 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 37775 = uvT +wtT b) We can encode polynomials in two variables x, y as a matrix so that the polynomial 3x2y xy2 + 5x3y3 is encoded by 26664 0 0 0 0 0 0 1 0 0 3 0 0 0 0 0 5 37775 Thus the i, j entry corresponds to the coe cient of xi 1yj 1. Say we have a polynomial p(x) = x+ 2x2 and polynomial q(y) = 5 y2 then we have p(x)q(y) encoded as26664 0 0 0 0 5 0 1 0 10 0 2 0 0 0 0 0 37775 = 26664 0 1 2 0 37775 [5 0 1 0 ] . Show that in general that a product of two polynomials p(x), q(y) can be encoded as uvT where u encodes p(x) and v encodes q(y). c) Can there exist polynomials p(x), q(y), r(x), s(y) (each of maximum degree 3) such that p(x)q(y) + r(x)s(y) = 1 + xy + x2y2?
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