A First Course in Linear Algebra · Problem C30

Linear Transformations — Robert Beezer

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

Problem

Define the linear transformation

T: \mathbb{C}^3 \to \mathbb{C}^2, \quad T\left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\right) = \begin{bmatrix} 2x_1 - x_2 + 5x_3 \\ -4x_1 + 2x_2 - 10x_3 \end{bmatrix}

Compute the preimages,

T^{-1}\left(\begin{bmatrix}2\\3\end{bmatrix}\right)

and

T^{-1}\left(\begin{bmatrix}4\\-8\end{bmatrix}\right)