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Inner Product
Spaces
7
...
3, 1
...
2, we saw that the dot product plays an essential role in
the discussion of lengths, distances, and projections in JR
...
If ideas such as projections are going to be used in these more
general spaces, it will be necessary to have a generalization of the dot product to
general vector spaces
...
3 leads
to the following definition
...
An inner product on V is a function ( , ) : V x V
Inner Product
such that
Inner Product Space
-4
JR
(1) (v, v) � 0 for all v EV and (v, v) = 0 if and only if v = 0
...
(symmetric)
(3) (v, sw + t z) = s(v, w) + t(v, z) for all s, t E JR and v, w, z EV
...
Remark
Every non-trivial finite-dimensional vector space V has in fact infinitely many different
inner products
...
EXAMPLE 1
The dot product on JR
...
EXAMPLE2
Show that the function ( , ) defined by
(x, y) = 2x1Y1 + 3x2y2
is an inner product on JR2
...
(x, x)= 2xi + 3x�
�
0 and (x, x)= 0 if and only if x =
0
...
2
...
Thus, it is symmetric
...
For any x,y,ZE lR
...
Thus, ( , ) is an inner product on IR2
...
See Problem Dl
...
Solution: We first verify that ( , ) satisfies the three properties of an inner product:
2
2
2
(1) (p, p) = (p(0)) + (p(1)) + (p(2)) � 0 for all p E P2
...
Thus ( , ) is positive definite
...
So, ( , ) is symmetric
...
Thus, ( , ) is an inner product on P2
...
In this inner product space, we have
2
2
(1 + x + x , 2 - 3x )= (1 + 0 + 0)(2 - 0 ) + (1 + 1 + 1)(2 - 3) + (1 + 2 + 4)(2 - 12)
= 2 - 3 - 70 = -71
EXAMPLE4
Let tr(C) represent the trace of a matrix (the sum of the diagonal entries)
...
If
A =
[�
_
�
]
[� �]
...
[� �]) ([� �][� -�D
([ � !])
and B =
then under this inner product, we have
=tr
= tr
EXERCISE 1
2
= 4 + 4= 8
Verify that (A, B)= tr(BT A) is an inner product for M(2, 2)
...
4?
Since these properties of the inner product mimic the properties of the dot product,
it makes sense to define the norm or length of a vector and the distance between vectors
in terms of the inner product
...
Then, for any v EV, we define the norm (or length)
�form
of v to be
Distance
llvll=
-J(v, v)
For any vectors v, w EV, the distance between v and w is
ll v-wll
Definition
A vector v in an inner product space V is called a
Unit Vector
EXAMPLES
Find the norm of A =
[� �]
unit vector if llvll = 1
...
Solution: We have
llAll =
-J(A, A)= -Jtr(AT A)= Ys+1= -y6
EXAMPLE6
Find the norm of p(x) = 1 - 2x - x2 in P2 under the inner product (p, q)
=
p(O)q(O)
+ p(l)q(l) + p(2)q(2)
...
Find the norm of p(x)
(p, q)
--j(p, p)
Find the norm of q(x)
(p, q) = p(-l)q(-1) +
=
x in P2 under the inner product
p(O)q(O) + p(l)q(l)
...
1 and 7
...
Hence, we extend this concept to general inner product spaces
...
Then two vectors
Orthogonal
are said to be orthogonal if
Orthonormal
be orthogonal if
have
(v;, v;)= 1
(v, w) = 0
...
The set of vectors
{v1,
...
The set is said to be orthonormal if we also
for all i
...
1 and 7
...
In particular, we
get that if '13 =
{v1,
...
If we have a basis
{w1,
, w11} for an inner
{v1,
...
EXAMPLE 7
Use the Gram-Schmidt Procedure to determine an orthonormal basis for § =
Span{l, x} of P
2
under the inner product ( p,q) = p(O)q(O) + p(l)q(l) + p(2)q(2)
...
Solution: Denote the basis vectors of § by p1(x)
1 and p2( x)
x
...
By using the Gram-Schmidt Procedure,
2
we take q1(x) = p1(x) = 1 and then let
=
=
Therefore, our orthogonal basis is {q1,q } = {1,x - 1}
...
x2 =
proJs
(x2,1)
1+
(x2, x- 1)
(x - 1)
llx-1112
!ili
0(1)+1(1)+4(1)
0(-1)+1(0)+4(1)
1+
(x - 1)
1 2+12+12
(-1 )2 +02+12
1
5
= -1+2(x - 1) = 2x - 3
3
=
PROBLEMS 7
...
Calculate the following
...
(i) Use the Gram-Schmidt Procedure to determine
an orthonormal basis for the following sub
spaces of M(2, 2)
...
[ ]
{[ � �],[� �],[� -�]}
{[� i],[� -�],[:1 �]}
4
...
1
_
A4 Define the inner product (1, y)
3
3x3y3 on IR
...
[-i]·[i]}
(b) Detemllne the coordinates of X
m
�
with re
AS Let {v1,
...
Prove that
spect to the orthogonal basis you found in (a)
...
Calculate the following
...
(a) (p,q) p(-l)q(-1) + p(O)q(O) + p(l)q(l)
=
(b) (p,q) p(O)q(O) + p(l)q(l) + p(3)q(3)
+ p(4)q(4)
(c) (p,q) p(-l)q(O) + p( l)q(l) + p(2)q(2)
+ p(3)q(3)
=
(ii) Use the orthogonal basis you found in part (i)
to determine proj5(1 + x + x2)
...
=
X1Y1 + 3x2 y2 +
(a) Use the Gram-Schmidt Procedure to determine
an orthogonal basis for
=
B3 On P define the inner product (p,q)
2
+ p(O)q(O) + p(l)q(l)
...
2
{[i]' nl, [i]}
[�]
(b) Determine the coordinates of X
=
with re
spect to the orthogonal basis you found in (a)
...
2 and sup
2
pose that ( ,) is an inner product on IR
...
2,
(1, st>
x1Y1
2
2
+ x Y1
2
2
2 2 2 2
=
(b) For the inner product in part (a), define a
matrix G, called the standard matrix of the in
ner product ('),by g;j (ei,ej) for i, j 1, 2
...
2
...
Show that G I
...
f 1 v1 +
...
Y2v2,
Conclusion
...
2, there exists a basis for JR
...
Moreover, when
1 and y are expressed in terms of this basis, (1, Y>
looks just like the standard inner product in JR
...
2
...
D2 Suppose that {v1, v2, v3} is a basis for an inner prod
uct space V with inner product ( , )
...
(c) Determine the matrix G of the inner product
(p , q )
p(O)q(O) + p( l )q ( l ) + p(2)q(2) for P2
with respect to the basis {1, x, x2}