ECE210a_lecture6_small

# ECE210a_lecture6_small - Inner product spaces Examples...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## This note was uploaded on 04/06/2010 for the course ECE 210a taught by Professor Chandrasekara during the Spring '08 term at UCSB.

### Page1 / 8

ECE210a_lecture6_small - Inner product spaces Examples...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online