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asst5 - y || 2 || x-y || 2 = 2 || x || 2 2 || y || 2 5 If U...

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MATH 235, Fall 2007 Assignment # 5 1. Compute quantities x · y and dist ( x , y ) for all choices of x from the set 1 - 1 0 , 1 - 1 + i 2 , 0 0 1 and y from the set 1 2 1 , 1 - 1 + 2 i i . 2. Find a basis for the subspace W of R 4 , where W = Span 1 2 1 0 , 1 3 1 0 , 2 3 2 1 . 3. Find the angle between the vectors x = (3 , 4) and y = (1 , - 7). 4. Verify the parallelogram law for vectors x , y in C n : ||
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Unformatted text preview: y || 2 + || x-y || 2 = 2 || x || 2 + 2 || y || 2 5. If U is a subspace of R n of dimension k , then prove that dim U ⊥ = n-k . Hint: use Theorem 3 from Chapter 6, The Rank Theorem (Theorem 14 from Chapter 2) and the fact that for any matrix rankA = rankA T . 6. If A is an n × n complex matrix then prove that ( A H x ) · y = x · ( A y ) for all x , y ∈ C n . 1...
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