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20f_03w_2be - L(They need not be a basis(c Find a matrix A...

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Math 20F Second Midterm March 3, 2003 PRINT NAME Write version on your blue book and VERSION B hand in this exam inside your blue book. There are a total of 50 points possible. ONE PAGE of notes is allowed. No calculators are allowed. You must show your work to receive credit. 1. (6 pts) Find a matrix T so that, if x R 3 has coordinates c in the basis v 1 = 0 1 0 v 2 = 0 0 - 1 v 3 = 2 0 0 , then it has coordinates T c in the standard basis i , j , k for R 3 . 2. (24 pts) A linear transformation from R 3 to R 3 is given by L ( x ) = x 1 + x 2 x 2 + x 3 x 1 - x 3 . (a) What is the kernel of L ? (b) Find a set of vectors that span the range of
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Unformatted text preview: L . (They need not be a basis.) (c) Find a matrix A such that L ( x ) = A x . (d) What is the dimension of the range of L ? Give a reason for your answer. 3. (10 pts) Suppose that A, B ∈ R n × n are nonsingular and that A and B are similar. Prove that A-1 and B-1 are similar. 4. (10 pts) Suppose V is a subspace of R n and W is a subspace of V . Prove that W ⊥ contains V ⊥ . WARNING: The final exam will probably not be in this room. END OF EXAM...
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