Chapter7 - Chapter 7: Inner-Product Vector Spaces Keith E....

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Unformatted text preview: Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Tarleton State University April 1, 2011 Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Overview Inner-Product Spaces Orthogonality Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Inner-Products Definition 1 An inner product on a vector space is an operation that creates, from any two vectors x and y , a scalar, denoted by h x , y i . These postulates must be fulfilled: 1. h x , x i > 0 whenever x 6 = . 2. h x , y i = h y , x i with h x , y i = h y , x i in the real case. 3. h x + y , z i = h x , z i + h y , z i 4. h x , y i = h x , y i , for any scalar . Definition 2 Let x , y R n . The standard inner product is defined by h x , y i = x 1 y 1 + + x n y n = x T y = y T x . For x , y C n , the standard inner product is h x , y i = x 1 y 1 + + x n y n = y H x , where y H is the Hermitian or conjugate transpose. Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Example Example 3 Let x = ( 1 , 2 , 3 , 4 ) and y = ( ,- 1 , 4 , 2 ) . Then h x , y i = 1 + 2 (- 1 ) + 3 4 + 4 2 = 18 . Remark 4 In R n , the inner product is the same as the dot product . h x , y i = x y . Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Other Inner-Products Definition 5 I In R n , choose w = ( w 1 ,..., w n ) where w i > , 1 i n are called weights . Then h x , y i = w 1 x 1 y 1 + + w n x n y n . I Consider C [ a , b ] (continuous real valued functions on [ a , b ] ). Then an inner product is h f , g i = b a f ( t ) g ( t ) dt . Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Inner-Product Properties Theorem 6 The following properties hold for any inner product. I h x , y i = h x , y i with h x , y i = h x , y i , in the real case. I h x , y + z i = h x , y i + h x , z i . I * m X i = 1 x i , y + = m X i = 1 h x i , y i . I * x , m X i = 1 y i + = m X i = 1 h x , y i i . I h x + y , x + y i = h x , x i + h x , y i + h y , x i + h y , y i with h x + y , x + y i = h x , x i + 2 h x , y i + h y , y i , in the real case. Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Norms on an Inner-Product Space Definition 7 Suppose a vector space V has an inner product, h , i . Then a norm can be defined by k x k = p h x , x i . Example 8 In R n with the standard inner product h x , x i = x 2 1 + + x 2 n , the induced norm is just the Euclidean distance: k x k = q x 2 1 + + x 2 n ....
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This note was uploaded on 01/16/2012 for the course MATH 332 taught by Professor Keithemmert during the Spring '11 term at Tarleton.

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Chapter7 - Chapter 7: Inner-Product Vector Spaces Keith E....

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