A First Course in Linear Algebra · Problem C31

Linear Transformations — Robert Beezer

Mathematics — Linear Algebra · pre-images-linear-transformations

Problem

For the linear transformation

S

compute the pre-images.

S: \mathbb{C}^3 \to \mathbb{C}^3, \quad S\left(\begin{bmatrix} a \\ b \\ c \end{bmatrix}\right) = \begin{bmatrix} a - 2b - c \\ 3a - b + 2c \\ a + b + 2c \end{bmatrix}

S^{-1}\left(\begin{bmatrix} -2 \\ 5 \\ 3 \end{bmatrix}\right) \qquad S^{-1}\left(\begin{bmatrix} -5 \\ 5 \\ 7 \end{bmatrix}\right)