A First Course in Linear Algebra · Problem C50

Representations — Robert Beezer

Mathematics — Linear Algebra · invertible-linear-transformations

Problem

Consider the linear transformation

S: M_{12} \to P_1

from the set of

1 \times 2

matrices to the set of polynomials of degree at most 1, defined by

S\left(\begin{bmatrix} a & b \end{bmatrix}\right) = (3a + b) + (5a + 2b)x

Prove that

S

is invertible. Then show that the linear transformation

R: P_1 \to M_{12}, \quad R(r + sx) = \begin{bmatrix} (2r - s) & (-5r + 3s) \end{bmatrix}

is the inverse of

S

, that is

S^{-1} = R