A First Course in Linear Algebra · Problem CRS.T45

Matrices — Robert Beezer

Mathematics — Linear Algebra · null-space-column-space-intersection

Problem

Suppose that

A

is an

m \times n

matrix and

B

is an

n \times m

matrix where

AB

is a nonsingular matrix. Prove that 1.

\mathcal{N}(B) = \{\mathbf{0}\}

\mathcal{C}(B) \cap \mathcal{N}(A) = \{\mathbf{0}\}

Discuss the case when

m = n

in connection with Theorem NPNT.