Chapter 3, Problem 9CTB

Let ${\text{x}}_{\text{1}}{\text{,x}}_{\text{2}}$ and ${\text{x}}_3$ be linearly independent vectors in ${ℝ}^{\text{4}}$ and let A be a nonsingular $4\times 4$ matrix. Prove that if
${\text{y}}_{\text{1}}=A{\text{x}}_1,{\text{y}}_2=A{\text{x}}_2,{\text{y}}_3=A{\text{x}}_3$
then ${\text{y}}_{\text{1}},{\text{y}}_2$ and ${\text{y}}_3$ are linearly independent.