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Midterm 2 Version A

Midterm 2 Version A - Answer Key to Second Midterm...

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Answer Key to Second Midterm Examination, Version 1 1. Let V be a two dimensional subspace of 5 . Answer the following questions (a correct answer without step by step description of your reasoning: maximum 10 points deduction each): (a) Find the dimension of V . The space V is the kernel of proj V . Since Im proj V V , rank proj V dim V 2. Thus the nullity is 5 2 3; so, the dimension of V is 3. (b) Prove that V V 0 . if x x 1 x 2 . . . x 5 is in V V , then x 2 1 x 2 2 x 2 3 x 2 4 x 2 5 x x 0 because the left x can be considered to be in V and the right x can be considered to be in V . Since x 2 j 0 for all j 1 2 5, the only possibility for x j to have x 2 1 x 2 2 x 2 5 0 is x 1 x 2 x 5 0. Thus x 0 and V V 0 . (c) For two distinct orthonormal basis v 1 v 2 and u 1 u 2 of V and any given vector x in 5 , let v v 1 x v 1 v 2 x v 2 and w u 1 x u 1 u 2 x u 2 . Prove that v w . Hint: Show first that x v is in V , and use this fact and (b) to show v w .
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