# D 0 find a basis and compute the dimension of s a3

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d = 0. Find a basis and compute the dimension of S . A–3. Let S and T be linear spaces and L : S T be a linear map. Say ~v 1 and ~v 2 are (distinct!) solutions of the equations L~x = ~ y 1 while ~w is a solution of L~x = ~ y 2 . Answer the following in terms of ~v 1 , ~v 2 , and ~w . a) Find some solution of L~x = 2 ~ y 1 - 7 ~ y 2 . b) Find another solution (other than ~w ) of L~x = ~ y 2 .

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Name (print) 3 A–4. Say you have matrices A and B . a) If A : R 7 R 7 , what are the possible dimensions of the kernel of A ? The image of A ? b) If B : R 3 R 5 , what are the possible dimensions of the kernel of B ? The image of B ? A–5. Give an example of 2 × 2 matrices A and B with AB = 0 but A 6 = 0 and B 6 = 0.
Name (print) 4 Part B Four questions, 15 points each (so 60 points total). B–1. Let C = 1 0 - 2 1 0 2 0 0 0 0 1 1 0 0 0 - 1 . [ NOTE: In this problem, there is no partial credit for incorrect computations.] a) Find the inverse of C . b) Find the inverse of C 2 .

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Name (print) 5 B–2. Define the linear maps A , B , and C from R 2 R 2 by the rules A rotates vectors by π/ 2 radians counterclockwise.

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