A First Course in Linear Algebra · Problem SLT-C40

Linear Transformations — Robert Beezer

Mathematics — Linear Algebra · surjective-linear-transformations

Problem

Show that the linear transformation

T

is not surjective by finding an element of the codomain,

\mathbf{v}

, such that there is no vector

\mathbf{u}

with

T(\mathbf{u}) = \mathbf{v}

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