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sample_tt1_2 - Math 235 Sample Midterm 2 NOTES Our Midterm...

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Math 235 Sample Midterm - 2 NOTES: - Our Midterm does not cover the Gram-Schmidt procedure. 1. Short Answer Problems a) Write a basis for the four fundamental subspaces of A = 1 0 0 - 1 0 0 1 1 0 0 0 0 . b) Determine if h p, q i = p ( - 1) q ( - 1) + p (1) q (1) is an inner product for P 2 . c) State the Rank-Nullity Theorem. d) Find the rank and nullity of the linear mapping T : P 2 M 2 × 2 ( R ) defined by T ( a + bx + cx 2 ) = c b 0 c 2. Let L : M 2 × 2 ( R ) M 2 × 2 ( R ) be given by L ( A ) = 1 2 3 4 A T . Find the matrix for L relative to the basis B of M 2 × 2 ( R ), where B = 1 0 0 0 , 0 1 0 0 , 0 0 1 0 , 0 0 0 1 3. Let B = { v 1 , v 2 } be a basis for V . Let a be a scalar constant. Let T : V V be linear and T ( v 1 ) = av 1 + av 2 , T ( v 2 ) = 3 v 1 - av 2 . For what values of a is T an isomorphism? 4. Let N be the plane with basis 1 0 - 1 , 1 - 1 1 . Define an explicit isomorphism to establish that P 1 and N are isomorphic. Prove that your map is an isomorphism. 5. Let T be a linear operator on an inner product space V , and suppose that h ~x, ~ y i = h T ( ~x ) , T ( ~ y ) i for all
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