Linear Algebra Done Right · Problem 3C.16

Linear Maps — Sheldon Axler

Mathematics — Linear Algebra · rank-matrices

Problem

Suppose

A

is an

m

-by-

n

matrix with

A \neq 0

. Prove that the rank of

A

is 1 if and only if there exist

(c_1, \ldots, c_m) \in \mathbf{F}^m

and

(d_1, \ldots, d_n) \in \mathbf{F}^n

such that

A_{j,k} = c_j d_k

for every

j = 1, \ldots, m

and every

k = 1, \ldots, n