1 MATH 21b Practice Questions Problem 1: Circle T if the given assertion is true, and circle F if it is false. There is no need to justify your answer. T F a) Let A = 1 2 12 1 21 !1 1" # $ $ % & ' ' . Then the equation A r x = 100! " # # $ % & & has no solution. T F b) Vectors r v 1, r v 2, r v 3, r v 4 in R4 are necessarily linearly independent if r v 4 is not equal to r v 1+ r v 2+ r v 3. T F c) If A is an n × n matrix and A = AA, then A must be either 0 or the identity matrix. T F d) Let A = 1 2 12 6 4!1 !4 !3" # $ $ % & ' ' . Then rref(A) = 1 0 !10 1 10 0 0" # $ $ % & ' ' . T F e) If a matrix has kernel = { r 0 }, then it must be invertible. T F f) The span of the rows of a matrix A must be the same as the span of the rows of rref(A). T F g) If A is a 2 × 2 matrix and AAA is the identity, then A must be the identity matrix. T F h) If A and B are n × n matrices and AB is invertible, then so are A and B. T F i) A linear transformation of R2 that sends 12! " # $ to 14! " # $ must send 14! " # $ to 18! " # $ . T F j) Let r v 1, r v 2 and r v 3 denote a given basis for R3 and let r v = - r v 1 - 3 r v 3 + 2 r v 2. Meanwhile, let T denote the linear transformation of R3 with matrix 1 2 12 6 4!1 !4 !3" # $ $ % & ' ' with respect to the basis { r v 1, r v 2, r v 3}. Then r v is in the image of T.2 Problem 2: Let r e 1 = 10! " # $ and r e 2 = 01! " # $ . Meanwhile, let r v 1 = 12! " # $ , r v 2 = !11" # $ % . This problem concerns the linear transformation, T: R2 → R2 that sends r v 1 to r e 1 and r v 2 to r e 2. a) Find the matrix of T with respect to the basis ( r e 1, r e 2). b) Find a vector r u such that T r u = r v 1. c) Find the matrix of T with respect to the basis ( r v 1, r v 2). d) Write down the matrix for T−1 with respect to the basis ( r v 1, r v 2). Problem 3: This problem concerns r v 1 = 3111! " # # # # $ % & & & & , r v 2 = 1!210" # $ $ $ $ % & ' ' ' ' , r v 3 = 11!11" # $ $ $ $ % & ' ' ' ' , r v 4 = !50!1!2" # $ $ $ $ % & ' ' ' ' , all vectors in R4. a) Write down a basis for the span of these four vectors. b) Let A denote the 4×4 matrix whose k’th column is r v k. Here, k ∈ {1, 2, 3, 4}. Give a basis for the kernel of A. c) Give a basis for the image of A. Problem 4: Let r v 1 = 122! " # # $ % & & , r v 2 = 330! " # # $ % & & . a) Write down an orthonormal basis for the span of { r v 1, r v 2}. b) Write down the matrix with respect to the standard basis of R3 for the linear transformation that gives the orthogonal projection to the span of { r v 1, r v 2}. c) Write down the matrix with respect to the standard basis of R3 for the linear transformation that gives the orthogonal projection to the orthogonal complement of the span of { r v 1, r v 2}. d) Give an orthogonal and invertible linear transformation of R3 that sends both of the vectors 100! " # # $ % & & and 010! " # # $ % & & to the span of { r v 1, r v 2}.3 Problem 5: This problem concerns the following five matrices: A = 1 1 !11 !2 01 1 1" # $ $ % & ' ' , B = 1 1 !21 !2 11 1 !2" # $ $ % & ' ' , C = 1 1 11 !2 1!2 0 2" # $ $ % & ' ' , Note that these matrices differ slightly one from the other. a) Compute AC b) Compute B + BT. Here, BT denotes the transpose of B. c) Which, if any, of the three matrices has columns that form a set { r a 1, r a 2, r a 3} such that r a 1 is orthogonal to r a 2 and both r a 1 and r a 2 are orthogonal to r a 3? d) Which, if any, of the three matricies can be written as r D where r is a real number and D is an orthogonal matrix? e) Which, if any, of the three matrices is not invertible? ANSWERS Problem 1: a) T b) F c) F d) T e) F f) T g) F h) T i) F j) T. Problem 2: a) 131 1!2 1" # $ % b) r u = !14" # $ % c) 131 1!2 1" # $ % d) 1 !12 1" # $ % Problem 3: a) { r v 1, r v 2, r v 3} b) 1111! " # # # # $ % & & & & c) { r v 1, r v 2, r v 3} . Problem 4: a) The vectors r u 1 = 13122! " # # $ % & & and r u 2 = 1321!2" # $ $ % & ' ' .4 b) A = r u 1 r u 1T + r u 2 r u 2T = 195 4 !24 5 2!2 2 8" # $ $ % & ' ' . c) A = 194 !4 2!4 4 !22 !2 1" # $ $ % & ' ' . d) A = 131 2 !22 1 22 !2 !1" # $ $ % & ' ' . Problem 5: a) 4 !1 0!1 5 !10 !1 4" # $ $ % & ' ' , b) = 2 2 !12 !4 2!1 2 !4" # $ $ % & ' ' …
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