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Unformatted text preview: 5.2 ORTHOGONAL SUBSPACES KIAM HEONG KWA pp. 214215 Two subspaces X,Y R n are said to be orthogonal , denoted X Orthogonal sub spaces Y , if h x , y i = x T y = 0 for all x X and all y Y . If X Y , then X Y = { } . To see this, let z X Y , so that k z k 2 = h z , z i = 0 and thus z = . p. 215 Let Y be a subspace of R n . The orthogonal complement of Y is Orthogonal com plement the set Y = { x R n h x , y i = x T y = 0 for all y Y } . Clearly, Y , so that Y 6 = . Let x 1 , x 2 Y and let , R . Then for any y Y , h x 1 + x 2 , y i = h x 1 , y i + h x 2 , y i = 0 , so that x 1 + x 2 Y . This shows that Y is a subspace of R n . Exercise 1. Prove that if Y R n is a subspace, then Y ( Y ) . Example 1 (Exercise 5.2.8 in the text) . If S = Span( x 1 , x 2 , , x k ) R n , then for any y R n , y S if and only if y x i for all i { 1 , 2 , ,k } . Proof. If y S , then y x i in view of the definition of S and the fact that x i S for all i { 1 , 2 , ,k } . Conversely, suppose that y x i for all i { 1 , 2 , ,k } . Let x S but otherwise arbitrary. Since S = Span( x 1 , x 2 , , x k ) , there are scalars c i s such that x = k i =1 c i x i . Hence h y , x i = * y , k X i =1 c i x i + = k X i =1 c i h y , x i i = k X i =1 c i 0 = 0 , indicating that y S . pp. 215216 Let A R m n . Recall that a vector w R m is in the column space of A if and only if A x = w for a vector x R n . Considering A as a map from R n into R m , it follows that such a vector b is in the Date : July 12, 2011. 1 2 KIAM HEONG KWA range R ( A ) of A . In other words, the column space of A is in fact the range of A as a map: R ( A ) = { w R m  A x = w for some x R n } = the column space of A. Likewise, the column space of A T is the range of A T as a map: R ( A T ) = { y R n  A T z = y for some z R m } = the column space of A T . Note that y R ( A T ) if and only if y T is in the row space of A . More important is the fact that Theorem 5.2.1: Fundamental Sub spaces Theorem N ( A ) = R ( A T ) and N ( A T ) = R ( A ) ....
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 Summer '08
 KIM
 Linear Algebra, Algebra

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