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Unformatted text preview: Inner product spaces Examples. Verify that the properties of the inner product hold in each case. 1. V = R n and ( x, y ) := x y = n summationdisplay i =1 x i y i . 2. We can define an inner product on R n using a given invertible matrix, A , by, ( x, y ) := x A Ay. 3. V = R m n , the space of m n matrices. Given A , B R m n , ( A, B ) := trace ( A T B ) (or trace ( A B ) for C m n ) 4. Complex valued functions: x ( w ) , y ( w ) C ( , ). ( x, y ) := integraldisplay x ( w ) y ( w ) dw. Roy Smith: ECE 210a: 6 .2 Inner product spaces Inner product spaces Defined for a pair of elements of a vector space, x, y X , ( x, y ) X : X X R (or possibly C ) . Defining properties: 1. ( x, x ) R , ( x, x ) and ( x, x ) = 0 x = 0. 2. ( x, y ) = ( x, y ) , for all scalars, . 3. ( x, y + z ) = ( x, y ) + ( x, z ) . 4. ( x, y ) = ( y, x ) . ( ( y, x ) denotes the complex conjugate). If the vector space is clear we will drop the explicit subscript. The pair, V , and ( , ) V are known as an inner product space . A complete inner product space is called a Hilbert space . Roy Smith: ECE 210a: 6 .1 Inner product spaces Compatible norm If we have an inner product space, we can define a compatible norm by, bardbl x bardbl := radicalbig ( x, x ) . This is not the only possible norm, but compatibility with the inner product is required to generalize R 3 intuition about distances and angles. Notice that this norm looks a lot like a Euclidean norm (or 2-norm). In R n it is the Euclidean norm. Norm properties: 1. bardbl x bardbl 0 and bardbl x bardbl = 0 x = 0 comes from the inner product properties. 2. bardbl x bardbl = ( x, x ) 1 / 2 = ( ( x, x ) ) 1 / 2 = ( | | 2 ( x, x ) ) 1 / 2 = | |bardbl x bardbl . 3. Triangle inequality. This one is trickier ... Roy Smith: ECE 210a: 6 .4 Inner product spaces Key idea: Inner products convey the idea of an angle between vectors. We will see that we can define such an angle by, cos = ( x, y ) bardbl x bardblbardbl y bardbl ....
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- Spring '08