Page 265 - DMTH502_LINEAR_ALGEBRA

Unit 24: Inner Product and Inner Product Spaces

Notes
Example 13: If F be the real field and V be the set of polynomials, in a variable x over F
of degree 2 or less. In V we define an inner product by: If p(x), q(x) V, then
1
(p(x), q(x)) = p ( ) ( ) dx
x
x
q
1
2
Let us start with the basis = 1, = x, = x of V and obtain orthogonal set by applying Gram-
1 2 3
Schmidt process. Let
1
1
=
1 2
1
1
2
as 1 = 1.dx 2.
1
’ = ( , )
2 2 2 1 1
1
1 1 1 x 2
= x x .1 dx x
2 1 2 2
1
So the orthonormal is given by
2
x x 3
= x
2 , 1 1/2 2
2 2
x dx
1
Finally
, = ( , ) ( , )
3 3 3 2 2 3 1 1
3 3 2 1 1
= x 2 x 2 , x x x , ,
2 2 2 2

Now
1
3 3 1 3 x 4
x 2 , x = x 2 ,xdx 0
2 2 1 2 4
1
and

1 1 1 1 x 3 1 2
x 2 , = x 2 ,1dx
2 2 1 2 3 3
1
Thus
1
, = x 2
3
3
and normalized is given by
3
x 2 1 3 x 2 1 3 10 2
, 2 1/2 (3x 1)
= 1 4 .
3 3 2 1
x dx
1 3

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