section46 - Chapter 4: The Vector Space R n MAT188H1F Lec03...

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Unformatted text preview: Chapter 4: The Vector Space R n MAT188H1F Lec03 Burbulla Chapter 4 Lecture Notes Fall 2007 Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n Chapter 4: The Vector Space R n 4.6: Projections and Approximation Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.6: Projections and Approximation Orthogonal Complements Definition: Let U be a subspace of R n . The set U = { X R n | X Y = 0 , for all Y U } is called the orthogonal complement of U . U is read U-perp. Theorem: U is a subspace of R n . Proof: 1. O Y = 0 is true for any Y ; so it is certainly true for all Y U . Thus O U . 2. Let X 1 and X 2 be in U ; let Y be in U . Then X 1 Y = 0 , X 2 Y = 0 ( X 1 + X 2 ) Y = X 1 Y + X 2 Y = 0 + 0 = 0 X 1 + X 2 U 3. Let X be in U ; let a be scalar; let Y be in U . Then X Y = 0 ( aX ) Y = a ( X Y ) = a 0 = 0 aX U Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.6: Projections and Approximation Example 1 Let U = { [ x y z ] T R 3 | x- 2 y + 3 z = 0 } . Then U = span n [1- 2 3] T o ; that is, U is the line normal to the plane, passing through the origin. Conversely, if V = span n [1- 2 3] T o , then V is the plane with equation x- 2 y + 3 z = 0; that is, the plane with normal vector [1- 2 3] T . This example illustrates the general property: if U is a subspace of R n , then U = U . Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.6: Projections and Approximation How To Find a Basis of U Suppose X 1 , X 2 , . . . , X k are k vectors in R n and U = span { X 1 , X 2 , . . . , X k } . Let Z = [ X 1 | X 2 | | X k ] be the n k matrix with columns X 1 , X 2 , . . . , X k . Then U = col Z = row Z T and U = null Z T , because X null Z T Z T X = 0 X T i X = 0 , for 1 i k X i X = 0 , for 1 i k X U Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.6: Projections and Approximation The Dimension of U Note, if r is the rank of the k n matrix Z T then dim U + dim U = dim(row Z T ) + dim(null Z T ) = r + ( n- r ) = n . Thus dim U = n- dim U ....
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section46 - Chapter 4: The Vector Space R n MAT188H1F Lec03...

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