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# Lecture7 - Lecture 7 7.1 Projection 1 Projection of b on sp...

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1 Lecture 7 7.1 Projection 1. Projection of b on sp( a ): a a b p cos Example: Find the projections of vector [1, 2] on x-axis, on the line y = 2x. Projection of b on a subspace W of R n : There are unique vectors W b and W b such that a) W b W; b) v b W , v W; c) W W b b b . Find w in W that minimizes w b :

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2 Let W = sp( k v v v ..., , , 2 1 ) be a subspace of R n . To find W from W: Let k v v v A 2 1 be an k n matrix. Then W = row space of A . The null space of A = null( A ) = 0 x A x is W . Example: Find W , if W = sp([2, -1, 1], [1, 0, -2]). Thm: Properties of W Let W be a subspace of R n . Then 1) W is a subspace. 2) dim(W ) = n dim(W); 3) The only vector common to W and W is 0 . 4) (W ) = W; 5) b R n , b can be expressed uniquely as W W b b b , where . , W b W b W W
3 Find the projection b on a subspace W of R n : 1) select a basis k v v v ..., , , 2 1 for W; 2) find a basis n k k v v v ..., , , 2 1 for W ; 3) let n n v r v r v r b ...

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