A First Course in Linear Algebra · Problem T40

Vector Spaces — Robert Beezer

Mathematics — Linear Algebra · matrix-equality-spanning-set

Problem

Prove the following variant of Theorem EMMVP that has a weaker hypothesis: Suppose that

C = \{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \ldots, \mathbf{u}_p\}

is a linearly independent spanning set for

\mathbb{C}^n

. Suppose also that

A

and

B

are

m \times n

matrices such that

A\mathbf{u}_i = B\mathbf{u}_i

for every

1 \leq i \leq n

. Then

A = B

. Can you weaken the hypothesis even further while still preserving the conclusion?