Exercise 5.2.15

Suppose $A$ is an $m\times n$-matrix and $\left\{\mathbf{w}_{1},\ldots,\mathbf{w}_{k}\right\}$ is a linearly independent set of vectors in $\mathbb{R}^{m}$. Now suppose $A\mathbf{z}_{i}=\mathbf{w}_{i}$. Show $\left\{\mathbf{z}_{1},\ldots,\mathbf{z}_{k}\right\}$ is also linearly independent.