Kunze Linear Algebra · Problem 3.4.6

Linear Transformations

Mathematics — Methods · rotation-operator-matrix-invertibility

Problem

Let

T

be the linear operator on

R^2

defined by

T(x_1, x_2) = (-x_2, x_1).

(a) What is the matrix of

T

in the standard ordered basis for

R^2

? (b) What is the matrix of

T

in the ordered basis

\mathfrak{B} = \{\alpha_1, \alpha_2\}

, where

\alpha_1 = (1, 2)

and

\alpha_2 = (1, -1)

? (c) Prove that for every real number

c

the operator

(T - cI)

is invertible. (d) Prove that if

\mathfrak{B}

is any ordered basis for

R^2

and

[T]_{\mathfrak{B}} = A

, then

A_{12}A_{21} \neq 0