Chapter 3.6, Problem 18E

Let A be a $5\times 3$ matrix of rank 3 and let $\left\{{\text{x}}_{\text{1}},{\text{x}}_2,{\text{x}}_3\right\}$ be a basis for ${ℝ}^{\text{3}}$ .
(a) Show that $N\left(A\right)=\left\{0\right\}$ .
(b) Show that if ${\text{y}}_{\text{1}}=A{\text{x}}_{\text{1}},{\text{y}}_2=A{\text{x}}_2$ , and ${\text{y}}_3=A{\text{x}}_3$ then ${\text{y}}_1{\text{,y}}_2$ and ${\text{y}}_3$ are linearly independent.
(c) Do the vectors ${\text{y}}_1{\text{,y}}_2,{\text{y}}_3$ from part (b) form a basis for ${\text{R}}^5$ ? Explain.