A First Course in Linear Algebra · Problem MR.T20

Representations — Robert Beezer

Mathematics — Linear Algebra · nonsingular-matrix-invertible-linear-transformation

Problem

Construct a new solution to Exercise B.T50 along the following outline. From the

n \times n

matrix

A

, construct the linear transformation

T: \mathbb{C}^n \to \mathbb{C}^n

T(\mathbf{x}) = A\mathbf{x}

. Use Theorem NI, Theorem IMILT and Theorem ILTIS to translate between the nonsingularity of

A

and the surjectivity/injectivity of

T

. Then apply Theorem ILTB and Theorem SLTB to connect these properties with bases.