Linear Algebra Done Right · Problem 3D.3

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · invertible-linear-maps

Problem

Suppose

V

is finite-dimensional and

T \in \mathcal{L}(V)

. Prove that the following are equivalent. (a)

T

is invertible. (b)

Tv_1, \ldots, Tv_n

is a basis of

V

for every basis

v_1, \ldots, v_n

V

. (c)

Tv_1, \ldots, Tv_n

is a basis of

V

for some basis

v_1, \ldots, v_n

V