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M346 First Midterm Exam, September 18, 2003 The exam is closed book, but you may have a single hand-written 8.5×11 crib sheet. There are 5 problems, each worth 20 points. The first four are calculational, and you MUST JUSTIFY YOUR ANSWERS. Part credit will be given, but answers without justification will not receive credit. The fifth problem is a series of true/false questions. For these, you do not have to show your work, and part credit will NOT be given. 1. Let M2,2 be the space of 2 × 2 real matrices, with standard basis 1 0 0 1 0 0 0 0 E ={ , , , } and alternate basis 0 0 0 0 1 0 0 1 1 2 0 1 0 0 0 0 3 2 B={ , , , }. Let A = . Compute 3 4 2 3 1 4 0 1 7 2 a) the change-of-basis matrix PEB , PEB = [b1 ]E [b2 ]E [b3 ]E 1 2 [b4 ]E = 3 4 0 1 2 3 b) the change-of-basis matrix PBE , −1 PBE = PEB 1 0 0 −2 1 0 = 1 −2 1 −2 5 −4 c) the coordinates of A in the E basis, and By inspection, [A]E = (3, 2, 7, 2)T . d) the coordinates of A in the B basis. [A]B = PBE [A]E = (3, −4, 6, −22)T . 1 0 0 0 1 0 0 1 4 0 0 0 1 1 2 2. Let Z = , and let L : M2,2 7→ M2,2 be given by L(M) = ZM, 3 4 where ZM is the matrix product of the 2 × 2 matrix Z and the 2 × 2 matrix M. L is a linear transformation. Find the matrix of L relative to the E basis of problem 1. 1 0 0 1 Since L(e1 ) = = e1 + 3e3 , L(e2 ) = = e2 + 3e4 , 3 0 0 3 2 0 0 2 L(e3 ) = = 2e1 + 4e3 , L(e4 ) = = 2e2 + 4e4 , we have 4 0 0 4 1 0 [L]E = 3 0 0 1 0 3 2 0 4 0 0 2 0 4 3. Let L : M2,2 7→ M2,2 be the linear transformation L a b c d = c a d b Compute a) the matrix of L in the E basis of problem 1, and Since L(e1 ) = e − 2, L(e2 ) = e4 , L(e3 ) = e1 , L(e4 ) = e3 , we have 0 1 [L]E = 0 0 0 0 0 1 1 0 0 0 0 0 1 0 b) the matrix of L in the B basis of problem 1. 3 2 1 0 −5 −4 −2 0 [L]B = PBE [L]E [L]B = 5 5 5 1 −15 −15 −18 −4 2 2 −5 −7 1 4. In R , let b1 = , b2 = ,v= ,y= . Let W1 be −1 3 5 0 the line x1 + 2x2 = 0, and let W2 be the line 3x1 + 5x2 = 0. a) Write v and y explicitly as linear combinations of b1 and b2 . v = 4b1 + 3b2 and y = 3b1 + b2 , as can be seen in each case by solving 2 equations in 2 unknowns. b) Viewing R2 as the (internal) direct sum of W1 and W2 , compute P1 v (the projection of v onto W1 ) and P2 y. Since W1 is the span of b1 and W2 is the span of b2 , this is just asking for the b1 part of v and the b2 part of y, namely P1 v = 4b1 and P2 y = b2 . c) Find the coordinates of v, P1 v, y and P2 y in the {b1 , b2 } basis. This is just restating the answers toparts (a) and (b), namely: 4 4 3 0 [v]B = , [P1 v]B = , [y]B = , [P2 y]B = . 3 0 1 1 5. True of False? Each question is worth 4 points. You do NOT need to justify your answers, and partial credit will NOT be given. a) The line 3x1 + 5x2 = 1 is a subspace of R2 . False. This line does not contain the origin. b) If A is a 3 × 5 matrix, then the dimension of the null space of A is at most 2. False. The dimension is at LEAST 2, and can be more (e.g., if A is the the zero matrix, then the null space is 5 dimensional). c) Every 5-dimensional subspace of R8 is the column space of a 8 × 5 matrix. True. Take 5 basis vectors for this subspace and write them as the columns of a 8 × 5 matrix. 1 1 4 3 1 5 , , and span a 3-dimensional subspace of R4 . d) The vectors 5 1 6 7 1 7 False. These vectors are linearly dependent and only span a 2-dimensional subspace of R4 . e) Let A be an n × m matrix. If there is only one solution to Ax = 0, then the columns of A are linearly independent. True. See Theorem 2.3. 2 3