q5rs - A ) ⊥ = n-2, and (Im A ) ⊥ = ker A T . Since...

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Quiz V 1. Use Gram-Schmidt to get a pair of orthonormal vectors with the same span as ⃗v 1 = 2 2 1 and ⃗v 2 = 1 1 5 Solution: Just following the Gram-Schmidt algorithm, ⃗u 1 = 1 | ⃗v 1 | ⃗v 1 = 2 / 3 2 / 3 1 / 3 and ⃗w 2 = ⃗v 2 - ( ⃗v 2 · ⃗u 1 ) ⃗u 1 = ⃗v 2 - 3 ⃗u 1 = - 1 - 1 4 . Finally, ⃗u 2 = 1 | ⃗w 2 | ⃗w 2 = 1 18 - 1 - 1 4 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 image of dimension 2, what is the dimension of the image of A T ? Solution: The dimension of (Im
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Unformatted text preview: A ) ⊥ = n-2, and (Im A ) ⊥ = ker A T . Since dim(ker A T ) + dim(Im A T ) = n , you get dim (Im A T ) = 2. This is an example of what follows from row rank equals column rank: the column rank of A is the dimension of Im(A), and the row rank is the dimension of Im( A T ). So row rank equals column rank implies that the dimensions of images are equal. [This is also Bretscher Theorem 5.3.9(c).]...
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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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