2.8 - 2.8 Subspaces of R n Definition: A subspace of R n is...

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Unformatted text preview: 2.8 Subspaces of R n Definition: A subspace of R n is a subset H R n such that 1. vector H 2. For all vectoru , vectorv H , vectoru + vectorv H Closed under addition 3. For all vectoru H ,c R , cvectoru H Closed under multiplication e.g. If vectorv 1 , ,vectorv m R n , then H = Span { vectorv 1 , vectorv m } is a subspace because 1. vector H as vector 0 = 0 vectorv 1 + + 0 vectorv m . 2. vectoru = c 1 vectorv 1 + + c m vectorv m vectorv = d 1 vectorv 1 + + d m vectorv m = vectoru + vectorv = ( c 1 + d 1 ) vectorv 1 + + ( c m + d m ) vectorv m H 3. cvectoru = cc 1 vectorv 1 + + cc m vectorv m H H is the subspace spanned by vectorv 1 , ,vectorv m . 1 EX: H = { ( x,y ) : x + y = 1 } R 2 . Is it a subspace? x + y = 1 y x What are subspaces of R 2 ? Recall that if vectorv 1 ,vectorv 2 R 2 are linearly independent, any vector vectorv R 2 is a linear combination of vectorv 1 and vectorv 2 , so Span { vectorv 1 ,vectorv 2 } = R 2 . If vectorv 1 = vectorv 2 , Span { vectorv 1 ,vectorv 2 } = { tvector v 2 : t R } So Span { vectorv 1 ,vectorv 2 } is the straight line through the origin in the direction of vectorv 2 ....
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This note was uploaded on 04/09/2011 for the course MATH 232 taught by Professor Russel during the Spring '10 term at Simon Fraser.

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2.8 - 2.8 Subspaces of R n Definition: A subspace of R n is...

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