Page 284 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes
Example 1: If V is an n-dimensional inner product space, then each ordered orthonormal
basis = { , …, } determines an isomorphism of V onto F with the standard inner product.
n
1 n
The isomorphism is simply
T (x + … + x ) = (x , …, x ).
1 1 n n 1 n
There is the superficially different isomorphism which determines of V onto the space F n×1
with (X|Y) = Y*X as inner product. The isomorphism is
[ ]

i.e., the transformation sending  into its coordinate matrix in the ordered basis . For any
ordered basis , this is a vector space isomorphism; however, it is an isomorphism of the two
inner product spaces if and only if is orthonormal.

Example 2: Here is a slightly less superficial isomorphism. Let W be the space of all
t
3 × 3 matrices A over R which are skew-symmetric, i.e., A = –A. We equip W with the inner
1 1
t
product (A|B) = tr (AB ), the being put in as a matter of convenience. Let V be the space R 3
2 2
with the standard inner product. Let T be the linear transformation from V into W defined by
0 x x
3 2
T (x , x , x ) = x 0 x
1 2 3 3 1
x 2 x 1 0
Then T maps V onto W, and putting

0 x x 0 y y
3 2 3 2
A = x 0 x , B = y 0 y
3 1 3 1
x 2 x 1 0 y 2 y 1 0
we have
t
tr (AB ) = x y + x y + x y + x y + x y
3 3 2 2 3 3 2 2 1 1
= 2 (x y + x y + x y ).
1 1 2 2 3 3
Thus ( | ) = (T |T ) and T is a vector space isomorphism. Note that T carries the standard basis
( , , ) onto the orthonormal basis consisting of the three matrices
1 2 3
0 0 0 0 0 1 0 1 0
0 0 1 , 0 0 0 , 1 0 0 .
0 1 0 1 0 0 0 0 0

Example 3: It is not always particularly convenient to describe an isomorphism in terms
of orthonormal bases. For example, suppose G = P*P where P is an invertible n × n matrix with
complex entries. Let V be the space of complex n × 1 matrices, with the inner product [X|Y] =
Y*GX.
Let W be the same vector space, with the standard inner product (X|Y) = Y*X. We know that V
and W are isomorphic inner product spaces. It would seem that the most convenient way to
describe an isomorphism between V and W is the following: Let T be the linear transformation
from V into W defined by T(X) = PX. Then
(TX|TY) = (PX|PY)

278 LOVELY PROFESSIONAL UNIVERSITY