A First Course in Linear Algebra · Problem M50

Representations — Robert Beezer

Mathematics — Linear Algebra · inverse-linear-transformation-basis-independence

Problem

Rework Example CIVLT, only in place of the basis

B

for

P_2

, choose instead to use the basis

C = \{1, 1+x, 1+x+x^2\}

. This will complicate writing a generic element of the domain of

T^{-1}

as a linear combination of the basis elements, and the algebra will be a bit messier, but in the end you should obtain the same formula for

T^{-1}

. The inverse linear transformation is what it is, and the choice of a particular basis should not influence the outcome.