A First Course in Linear Algebra · Problem B.T50

Vector Spaces — Robert Beezer

Mathematics — Linear Algebra · nonsingular-matrices-bases

Problem

Theorem UMCOB says that unitary matrices are characterized as those matrices that ``carry'' orthonormal bases to orthonormal bases. This problem asks you to prove a similar result: nonsingular matrices are characterized as those matrices that ``carry'' bases to bases. More precisely, suppose that

A

is a square matrix of size

n

and

B = \{\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3, \ldots, \mathbf{x}_n\}

is a basis of

\mathbb{C}^n

. Prove that

A

is nonsingular if and only if

C = \{A\mathbf{x}_1, A\mathbf{x}_2, A\mathbf{x}_3, \ldots, A\mathbf{x}_n\}

is a basis of

\mathbb{C}^n

. (See also Exercise PD.T33, Exercise MR.T20.)

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