Published on 21 Apr 2013

Department

Mathematics

Course

MATB24H3

Professor

University of Toronto at Scarborough

Department of Computer & Mathematical Sciences

MAT B24S Fall 2011

MAT B24

Lecture 11.

Inner-Product Spaces

1 Inner-Product Spaces

Review from MATA23: For all v,wand u∈Rnwe have v·uis the real

number given by:

v·u=v1u1+v2u2+···+vnun.

We also know that the following properties are satisﬁed: If v,u,w∈

Rnand r∈R:

1. v·u=u·v

2. u·(v+w) = u·v+u·w

3. r(v·u) = (rv)·u=v·(ru)

4. v·v≧0, and v·v= 0 if and only if v=0.

On other vector spaces we may deﬁne an analogous product that satisﬁes the

above properties and call it inner product. We will use the notation <v,u>

for the inner product of the vectors v,u.

DEFINITION: An inner product on a real vector space Vis a function

that associates a real number <v,u>with each pair of vectors vand uin

Vsuch that the following are satisﬁed ∀v,u,w∈Vand scalars r:

1

1. <v,u>=<u,v>Symmetry.

2. <u+v,w>=<u,w>+<v,w>Additivity.

3. < rv,u>=<v, ru>=r < v,u>Homogeneity

4. <v,v>≧0 and <v,v>= 0 if and only if v=0

A vector space together with an inner product deﬁned on it is called an inner

product space.

EXAMPLE: In the vector space R2verify that

<v,u>= 3v1u1+ 2v2u2

for all vectors v,u∈R2is an inner product.

Solution: Let v= [v1, v2], u= [u1, u2] and w= [w1, w2] and r∈R

1. <v,u>= 3v1u1+ 2v2u2= 3u1v1+ 2u2v2=<u,v>.

2. <v+u,w>= 3(v1+u1)w1+2(v2+u2)w2= 3v1w1+3u1w1+2v2w2+

2u2w2= (3v1w1+ 2v2w2)+(3u1w1+2u2w2) = <v,w>+<u,w>.

3. < ru,v>= 3ru1v1 + 2ru2v2=<u, rv>=r < u,v>.

4. <v,v>= 3v2

1+2v2

2≥0 and it equals to 0 if and only if v1=v2= 0

or v= [0,0].

As with dot product

1. we deﬁne the norm of v∈Vby

kvk=√<v,v>

and the distance between two non zero vectors uand vby d(u,v) =

ku−vk,

2. we have that

cos θ=<u,v>

kukkvk

where θis the angle between uand v

2

3. the Cauchy-Schwarz Inequality:

|<u,v>| ≤ kukkvk

holds,

4. the Triangle Inequality

kv+uk ≤ kvk+kuk

also holds and

5. we say that u,vare orthogonal if and only if <u,v>= 0

Cauchy-Schwarz Inequality: |<v,w>| ≤ kvkkwk

proof. We have that for all scalars r and s,

0≤ krv+swk2=< rv+sw, rv+sw>

=r2<v,v>+rs < v,w>+sr < w,v>+s2<w,w>

=r2<v,v>+2rs < v,w>+s2<w,w> .

Choose r=<w,w>

s=−<v,w>

=<w,w>2<v,v>−2<w,w> < w,v>2+<v,w>2<w,w>

=<w,w>2<v,v>−<w,w> < w,v>2

=<w,w>[<w,w> < v,v>−<v,w>2]

∴<w,w>[<w,w> < v,v>−<v,w>2]≥0.

Since <w,w>≥0 then kwk2kvk2−<v,w>2≥0 or <v,w>2≤

kwk2kvk2or |<v,w>| ≤ kwkkvk.

EXAMPLE: Let C[a, b] be the set of all continuous function on [a, b]. Show

that < f, g > =Zb

a

f(x)g(x)dx is an inner product on C[a, b].

(Norm in C[a, b] is:

kfk=< f, f >1/2=sZb

a

f2dx )

3

## Document Summary

Review from mata23: for all v, w and u rn we have v u is the real number given by: v u = v1u1 + v2u2 + + vnun. We also know that the following properties are satis ed: if v, u, w . On other vector spaces we may de ne an analogous product that satis es the above properties and call it inner product. We will use the notation < v, u > for the inner product of the vectors v, u. Definition: an inner product on a real vector space v is a function that associates a real number < v, u > with each pair of vectors v and u in. V such that the following are satis ed v, u, w v and scalars r : A vector space together with an inner product de ned on it is called an inner product space. Example: in the vector space r2 verify that.