Page 127 - DMTH502_LINEAR_ALGEBRA

Unit 8: Isomorphism

Notes
n×n
Example 4: Let F denote the set of n × n matrices with entries in a field F. This set is a
2
vector space over F and it is isomorphic to the space of column vectors of length n .
Self Assessment

mn
1. Show that F m×n is isomorphic to F .
2. Let V be the set of complex numbers regarded as a vector space over the field of real
numbers. Define a function T from V into the space of 2 × 2 real matrices, as follows. If
z x iy with x and y real numbers, then

x 7y 5y
T z .
10y x 7y

(a) Verify that T z z T z T z
1 2 1 2
(b) Verify that T is a one-one (real) linear transformation of V into the space of 2×2 real
matrices.

8.2 Summary

 A homomorphism is a mapping T of the space V into W over the same field F, preserving
all the algebraic structures of the system. If T, in addition is one-to-one we call the mapping
an isomorphism.
 Two spaces V and W are isomorphic only if the dim V = dim W.

8.3 Keywords

Isomorphism: T is an isomorphism of V into W over the same field F if T transforms a subset S
of independent vectors into T(S) a set of independent vectors of W.
Transformation: A transformation T of the space V into W is isomorphic if T is a non-singular
transformation.

8.4 Review Questions

1. Let U and V be finite dimensional vector space over the field F. Prove that U and V are
isomorphic if and only if dim U = dim V.
2. Let V and W be vector spaces over the field F and let T be an isomorphism of V onto W.
1
1
Prove that T T T T is an isomorphism of L V ,V onto L W ,W .
2 1 2 1
8.5 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra
I.N. Herstein, Topics in Algebra

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