A First Course in Linear Algebra · Problem ILT-C31

Linear Transformations — Robert Beezer

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

Problem

Given that the linear transformation

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

T\left(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\right) = \begin{bmatrix} 2x + y \\ 2y + z \\ x + 2z \end{bmatrix}

is injective, show directly that

\{T(\mathbf{e}_1), T(\mathbf{e}_2), T(\mathbf{e}_3)\}

is a linearly independent set.