A First Course in Linear Algebra · Problem SLT-T20

Linear Transformations — Robert Beezer

Mathematics — Linear Algebra · range-column-space

Problem

Suppose that

A

is an

m \times n

matrix. Define the linear transformation

T

T: \mathbb{C}^n \to \mathbb{C}^m, \quad T(\mathbf{x}) = A\mathbf{x}

Prove that the range of

T

equals the column space of

A

\mathcal{R}(T) = \mathcal{C}(A)