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sample_tt1_2 - Math 235 1 Short Answer Problems Sample Term...

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Math 235 Sample Term Test 1 - 2 1. Short Answer Problems a) Write a basis for the rowspace, columnspace and nullspace of A = 1 0 0 - 1 0 0 1 1 0 0 0 0 . b) Let B = { ~v 1 ,...,~v n } be orthonormal in an inner product space V and let ~v = a 1 ~v 1 + ··· + a n ~v n . Prove that a i = < ~v,~v i > . c) State the Rank-Nullity Theorem. d) Find the rank and nullity of the linear mapping T : P 2 M (2 , 2) defined by T ( a + bx + cx 2 ) = ± c b 0 c ² . 2. Let L : M (2 , 2) M (2 , 2) be given by L ( A ) = ± 1 2 3 4 ² A T . Find the matrix for L relative to the standard basis B of M (2 , 2), where B = ³± 1 0 0 0 ² , ± 0 1 0 0 ² , ± 0 0 1 0 ² , ± 0 0 0 1 ²´ . 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
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sample_tt1_2 - Math 235 1 Short Answer Problems Sample Term...

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