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gramschmidt - 642:527 The Gram-Schmidt Algorithm The...

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642:527 The Gram-Schmidt Algorithm The context of this discussion is a space of vectors V with an inner product denoted h v , w i . The norm associated to the inner product is denoted by k v k and is defined by k v k = q h v , v i . Orthogonality and orthonormality are defined relative to the inner product , ·i ; that is u and w are orthogonal if h u , w i = 0 and they are orthonormal if they are orthogonal and k u k =1, k w k =1. The problem we address is this: given a set { v 1 , . . . , v k } of k linearly independent vectors in V , find an orthonormal set of vectors { ˆ e 1 , . . . , ˆ e k } such that for every j , 1 j k , Span { ˆ e 1 , . . . , ˆ e j } = Span { v 1 , . . . , v j } . for all j = 1 , 2 , . . . , k . (1) Recall that this equality of spans means any linear combination of the vectors v 1 , . . . , v j can be written as a linear combination of the vectors ˆ e 1 , . . . , ˆ e j and vice-versa. To understand the Gram-Schmidt algorithm one needs to remember the following basic fact. If { u 1 , . . . , u n } is a set of orthogonal vectors and v is another vector, then n X i =1 h v , u i i h u i , u 1 i u i is the projection of v onto Span { u 1 , . . . , u n } , and (2) v - n X i =1 h v , u i i h u i , u 1 i u i is orthogonal to every vector in Span { u 1 , . . . , u n } . (3) These two statements are really the same, because the projection of v onto Span { u 1 , . . . , u k } is defined precisely by condition (3). Here is the algorithm for a given set v 1 , . . . , v k of linearly independent vectors.
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