Linear Algebra Done Right · Problem 3D.16

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · matrix-multiplication

Problem

Prove that every linear map from

\mathbf{F}^{n,1}

\mathbf{F}^{m,1}

is given by a matrix multiplication. In other words, prove that if

T \in \mathcal{L}(\mathbf{F}^{n,1}, \mathbf{F}^{m,1})

, then there exists an

m

-by-

n

matrix

A

such that

Tx = Ax

for every

x \in \mathbf{F}^{n,1}