Page 257 - DMTH502_LINEAR_ALGEBRA

Unit 24: Inner Product and Inner Product Spaces

suppose (|) is an inner product on W. If T is a non-singular linear transformation from V into W, Notes
then the equation
pr ( , ) = (T |T )

defines an inner product pr on V. The inner product in Example 4 is a special case of this
situation. The following are also special cases.
(a) Let V be a finite-dimensional vector space, and let,

 = { ..., )
1 n
n
be an ordered basis for V. Let , ..., be the standard basis vectors in F , and let T be the
1 n
n
linear transformation from V into F such that T = , j = 1, ..., n. In other words, let T be
j j
n
the ‘natural’ isomorphism of V onto F that is determined by . If we take the standard
n
inner product on F , then
n
pr x , y x y .
j j k k j j
j k j 1
Thus, for any basis for V there is an inner product on V with the property ( | ) = ; in
j k jk
fact, it is easy to show that there is exactly one such inner product. Later we shall show that
every inner product on V is determined by some basis  in the above manner.
(b) We look again at Example 5 and take V = W, the space of continuous functions on the unit
interval. Let T be the linear operator ‘multiplication by t,’ that is, (Tf) (t) = tf(t), 0 t 1. It
is easy to see that T is linear. Also T is non-singular; for suppose Tf = 0. Then tf(t) = 0 for
0 t 1; hence f(t) = 0 for t > 0. Since f is continuous, we have f(0) = 0 as well, or f = 0. Now
using the inner product of Example 5, we construct a new inner product on V by setting
1
pr(f,g) = (Tf )( )(Tg )( )dt
t
t
0
1
2
= f ( ) ( )t dt .
g
t
t
0
We turn now to some general observations about inner products. Suppose V is complex vector
space with an inner product. Then for all , in V
( | ) = Re ( | ) + i Im ( | ) ... (1)
where Re ( | ) and Im ( | ) are the real and imaginary parts of the complex number ( | ). If
z is a complex number, then Im (z) = Re (– iz). It follows that
Im ( | ) = Re [– i( | )] = Re ( |i ).
Thus the inner product is completely determined by its ‘real part’ in accordance with

( | ) = Re ( | ) + i Re ( |i ) ... (2)
Occasionally it is very useful to know that an inner product on a real or complex vector space is
determined by another function, the so-called quadratic form determined by the inner product.
To define it, we first denote the positive square root of ( | ) by|| ||; || || is called the norm
1
of with respect to the inner product. By looking at the standard inner products in R , C , R , and
1
2
3
R , the reader, should be able to convince himself that it is appropriate to think of the norm of
as the ‘length’ or ‘magnitude’ of . The quadratic form determined by the inner product is the
function that assigns to reach vector the scalar|| || . It follows from the properties of the
2
inner product that
2
( ) = 2 2 Re ( | ) 2

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