Exercise 19

Where $f\colon V\to W$ is linear, suppose that $f(\vec{v}_{1})=\vec{w}_{1}$, …, $f(\vec{v}_{n})=\vec{w}_{n}$ for some vectors $\vec{w}_{1}$, …, $\vec{w}_{n}$ from $W$.

• If the set of $\vec{w}\,$’s is independent, must the set of $\vec{v}\,$’s also be independent?

• If the set of $\vec{v}\,$’s is independent, must the set of $\vec{w}\,$’s also be independent?

• If the set of $\vec{w}\,$’s spans $W$, must the set of $\vec{v}\,$’s span $V$?

• If the set of $\vec{v}\,$’s spans $V$, must the set of $\vec{w}\,$’s span $W$?