A First Course in Linear Algebra · Problem ILT-C20

Linear Transformations — Robert Beezer

Mathematics — Linear Algebra · injective-linear-transformations

Problem

The linear transformation

T: \mathbb{C}^4 \to \mathbb{C}^3

is not injective. Find two inputs

\mathbf{x}, \mathbf{y} \in \mathbb{C}^4

that yield the same output (that is

T(\mathbf{x}) = T(\mathbf{y})

T\left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = \begin{bmatrix} 2x_1 + x_2 + x_3 \\ -x_1 + 3x_2 + x_3 - x_4 \\ 3x_1 + x_2 + 2x_3 - 2x_4 \end{bmatrix}