q5ts - A T Solution The dimension of(ker A ⊥ = m-2...

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Quiz V 1. Use Gram-Schmidt to get a pair of orthonormal vectors with the same span as ⃗v 1 = 1 1 1 1 and ⃗v 2 = 6 4 6 4 Solution: Just following the Gram-Schmidt algorithm, ⃗u 1 = 1 | ⃗v 1 | ⃗v 1 = 1 / 2 1 / 2 1 / 2 1 / 2 and ⃗w 2 = ⃗v 2 - ( ⃗v 2 · ⃗u 1 ) ⃗u 1 = ⃗v 2 - 10 ⃗u 1 = 1 - 1 1 - 1 . Finally, ⃗u 2 = 1 | ⃗w 2 | ⃗w 2 = 1 / 2 - 1 / 2 1 / 2 - 1 / 2 The set { ⃗u 1 ,⃗u 2 } is orthonormal and has the same span as { ⃗v 1 ,⃗v 2 } . 2. If A is an n × m matrix (m columns and n rows) with a kernel of dimension 2, what is the dimension of the kernel of
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Unformatted text preview: A T ? Solution: The dimension of (ker A ) ⊥ = m-2, and (ker A ) ⊥ = Im A T . Since dim(ker A T ) + dim(Im A T ) = n , you get dim (ker A T ) = n-m + 2. There is a puzzle here: why can’t n-m+2 be negative? The answer is that, if the kernel of A has dimension 2, then the image of A has dimension m-2, and that is a subspace of R n . So you must have n ≥ m-2....
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This note was uploaded on 02/06/2011 for the course MATH 33a taught by Professor Lee during the Fall '08 term at UCLA.

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