A First Course in Linear Algebra · Problem ILT-C32

Linear Transformations — Robert Beezer

Mathematics — Linear Algebra · injective-linear-transformations-linear-independence

Problem

Given that the linear transformation

T: \mathbb{C}^2 \to \mathbb{C}^3

T\begin{pmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \end{pmatrix} = \begin{bmatrix} x + y \\ 2x + y \\ x + 2y \end{bmatrix}

is injective, show directly that

\{T(\mathbf{e}_1), T(\mathbf{e}_2)\}

is a linearly independent set.