# 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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## This note was uploaded on 06/22/2008 for the course ENGINEERIN MAT188 taught by Professor Burbula during the Fall '07 term at University of Toronto.

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

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