Page 259 - DMTH502_LINEAR_ALGEBRA

Unit 24: Inner Product and Inner Product Spaces

Notes
x G x 0
j jk k .... (7)
, j k
From this we see immediately that each diagonal entry of G must be positive; however, this
condition on the diagonal entries is by no means sufficient to insure the validity of (6). Sufficient
conditions for the validity of (6) will be given later.

The above process is reversible; that is, if G is any n n matrix over F which satisfies (6) and the
condition G = G*, then G is the matrix in the ordered basis  of an inner product on V. This inner
product is given by the equation
( | ) = Y*GX
where X and Y are the coordinate matrices of and in the ordered basis .

Self Assessment

1. Let V be a vector space (|) an inner product on V.

(a) Show that (o| ) = 0 for all in V.
(b) Show that if ( | ) = 0 for all in V, then = 0.
2. Let (|) be the standard inner product on R . 2
(a) Let = (1, 2), = (– 1, 1). If is a vector such that ( | ) = –1 and ( | ) = 3, find .
(b) Show that for any in R we have
2
= ( | ) + ( | )
2 1 2 2
Where = (1, 0) and = (0,1).
1 2
24.2 Inner Product Space

After gaining some insight about an inner product we want to see how to combine a vector space
to some particular inner product in it. We shall thereby establish the basic properties of the
concept of length and orthogonality which are imposed on the space by the inner product.
Definition: An Inner Product space is a real or complex vector space together with a specified
inner product on that space.
A finite-dimensional real inner product space is often called a Euclidean Space. A complex inner
product space is often referred to as a unitary space.
We now introduce the theorem:
Theorem 1. If V is an inner product space, then for any , in V and any scalar

(i) c c ;

(ii) 0 for 0;

(iii) ( | )

(iv)

Proof: Statements (i), (ii) can be proved from various definitions. The inequality in (iii) is valid
for = 0. If 0, put

LOVELY PROFESSIONAL UNIVERSITY 253