A First Course in Linear Algebra · Problem SLT-C22

Linear Transformations — Robert Beezer

Mathematics — Linear Algebra · surjective-linear-transformations

Problem

The linear transformation

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

is not surjective. Find an output

\mathbf{w} \in \mathbb{C}^3

that has an empty pre-image (that is

S^{-1}(\mathbf{w}) = \emptyset

S\left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\right) = \begin{bmatrix} 2x_1 + x_2 + 3x_3 - 4x_4 \\ x_1 + 3x_2 + 4x_3 + 3x_4 \\ -x_1 + 2x_2 + x_3 + 7x_4 \end{bmatrix}