Linear Algebra Done Right · Problem 3C.17

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · rank-matrices

Problem

Suppose

T \in \mathcal{L}(V)

, and

u_1, \ldots, u_n

and

v_1, \ldots, v_n

are bases of

V

. Prove that the following are equivalent. (a)

T

is injective. (b) The columns of

\mathcal{M}(T)

are linearly independent in

\mathbf{F}^{n,1}

. (c) The columns of

\mathcal{M}(T)

span

\mathbf{F}^{n,1}

. (d) The rows of

\mathcal{M}(T)

span

\mathbf{F}^{1,n}

. (e) The rows of

\mathcal{M}(T)

are linearly independent in

\mathbf{F}^{1,n}

. Here

\mathcal{M}(T)

means

\mathcal{M}(T, (u_1, \ldots, u_n), (v_1, \ldots, v_n))