# Lecture11.pdf

106 views9 pages
School
UTSC
Department
Mathematics
Course
MATB24H3 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 uRnwe 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 rR:
1. v·u=u·v
2. u·(v+w) = u·v+u·w
3. r(v·u) = (rv)·u=v·(ru)
4. v·v0, 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,wVand scalars r:
1
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 9 pages and 3 million more documents. 1. <v,u>=<u,v>Symmetry.
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,uR2is an inner product.
Solution: Let v= [v1, v2], u= [u1, u2] and w= [w1, w2] and rR
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
20 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 vVby
kvk=<v,v>
and the distance between two non zero vectors uand vby d(u,v) =
kuvk,
2. we have that
cos θ=<u,v>
kukkvk
where θis the angle between uand v
2
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 9 pages and 3 million more documents. 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>20 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
Unlock document

This preview shows pages 1-3 of the document.
Unlock all 9 pages and 3 million more documents.

## 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.