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Chapter7

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

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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

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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 = 0 . 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 α x , y i = α h x , y i , for any scalar α . 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.

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Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Example Example 3 Let x = ( 1 , 2 , 3 , 4 ) and y = ( 0 , - 1 , 4 , 2 ) . Then h x , y i = 1 · 0 + 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 > 0 , 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 = ˆ b a f ( t ) g ( t ) dt .

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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 , α y i = α h x , y i with h x , α y i = α 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, , ·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 . Of course, ANY inner product can be used to form a norm!

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Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Normalized Vectors Definition 9 A normalized vector x is defined to be x k x k = 1 k x k x . Example 10 Note that the length of a normalized vector is always equal to one.
Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Norm Theory Theorem 11 In any inner-product space, the norm has these properties: I k x k > 0 for every nonzero vector x .

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

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