16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 3.4.7

Linear Transformations

Mathematics — Linear Algebra · matrix-representation-change-of-basis

Problem

Let

T

be the linear operator on

R^3

defined by

T(x_1, x_2, x_3) = (3x_1 + x_3,\; -2x_1 + x_2,\; -x_1 + 2x_2 + 4x_3).

(a) What is the matrix of

T

in the standard ordered basis for

R^3

? (b) What is the matrix of

T

in the ordered basis

\{\alpha_1, \alpha_2, \alpha_3\}

where

\alpha_1 = (1, 0, 1)

\alpha_2 = (-1, 2, 1)

, and

\alpha_3 = (2, 1, 1)

? (c) Prove that

T

is invertible and give a rule for

T^{-1}

like the one which defines

T