Abstract Algebra: Theory and Applications · Problem 20.16

Fields — Thomas Judson

Mathematics — Abstract Algebra · vector-space-isomorphism

Problem

Let

V

and

W

be finite dimensional vector spaces of dimension

n

over a field

F

. Suppose that

T : V \to W

is a vector space isomorphism. If

\{v_1, \ldots, v_n\}

is a basis of

V

, show that

\{T(v_1), \ldots, T(v_n)\}

is a basis of

W

. Conclude that any vector space over a field

F

of dimension

n

is isomorphic to

F^n