A First Course in Linear Algebra · Problem CRS.T40

Matrices — Robert Beezer

Mathematics — Linear Algebra · column-space-subset

Problem

Suppose that

A

is an

m \times n

matrix and

B

is an

n \times p

matrix. Prove that the column space of

AB

is a subset of the column space of

A

, that is

\mathcal{C}(AB) \subseteq \mathcal{C}(A)

. Provide an example where the opposite is false, in other words give an example where

\mathcal{C}(A) \not\subseteq \mathcal{C}(AB)