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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, αy ) = α ( x, y ) , for all scalars, α . 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 αx bardbl = ( αx, αx ) 1 / 2 = ( α α ( x, x ) ) 1 / 2 = ( | α | 2 ( x, x ) ) 1 / 2 = | α |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 θ = ( x, y ) bardbl x bardblbardbl y bardbl ....
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- Spring '08
- Linear Algebra, Hilbert space, inner product, Inner product space, Roy Smith