Chapter7

# 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 January 15, 2010 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 ( x , y ) . These postulates must be fulfilled: 1. ( x , x ) > 0 whenever x negationslash = . 2. ( x , y ) = ( y , x ) with ( x , y ) = ( y , x ) in the real case. 3. ( x + y , z ) = ( x , z ) + ( y , z ) 4. ( x , y ) = ( x , y ) , for any scalar . Definition 2 Let x , y R n . The standard inner product is defined by ( x , y ) = x 1 y 1 + + x n y n = x T y = y T x . For x , y C n , the standard inner product is ( x , y ) = 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 ( x , y ) = 1 + 2 ( 1 ) + 3 4 + 4 2 = 18 . Remark 4 In R n , the inner product is the same as the dot product . ( x , y ) = x y . Chapter 7: Inner-Product Vector Spaces Keith E. Emmert Inner-Product Spaces Orthogonality Other Inner-Products Definition 5 In R n , choose w = ( w 1 , . . . , w n ) where w i > , 1 i n are called weights . Then ( x , y ) = w 1 x 1 y 1 + + w n x n y n . Consider C [ a , b ] (continuous real valued functions on [ a , b ] ). Then an inner product is ( f , g ) = 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. ( x , y ) = ( x , y ) with ( x , y ) = ( x , y ) , in the real case. ( x , y + z ) = ( x , y ) + ( x , z ) . (Bigg m summationdisplay i = 1 x i , y )Bigg = m summationdisplay i = 1 ( x i , y ) . (Bigg x , m summationdisplay i = 1 y i )Bigg = m summationdisplay i = 1 ( x , y i ) . ( x + y , x + y ) = ( x , x ) + ( x , y ) + ( y , x ) + ( y , y ) with ( x + y , x + y ) = ( x , x ) + 2 ( x , y ) + ( y , y ) , 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, ( , ) . Then a norm can be defined by bardbl x bardbl = radicalbig ( x , x ) ....
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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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