# The kernel of $\bigwedge^r(T): \bigwedge^r(W)\to \bigwedge^r(V)$?

The full problem is:

Given $$T: W\to V$$, a linear transformation of $$F$$-vector spaces, such that $$\text{ker} T = 0$$, show that the kernel of $$\bigwedge^r(T): \bigwedge^r(W)\to \bigwedge^r(V)$$ is also zero.

I'm confused on how to deal with this problem when $$T$$ is not an endomorphism. It would also be helpful to know what $$\bigwedge^r(T)$$ is supposed to look like.

Thanks

• HINT: If $\ker T = \{0\}$, then $T$ maps sets of $r$ linearly independent vectors to linearly independent vectors. – Ted Shifrin Mar 19 at 23:32

Hint: Use the fact that a linear transformation $$T \colon W \rightarrow V$$ is injective if and only if there exists a linear transformation $$S \colon V \rightarrow W$$ such that $$S \circ T = \operatorname{id}_{W}$$. Then use the functoriality of $$\Lambda$$ (that is, $$\Lambda^r(S \circ T) = \Lambda^r(S) \circ \Lambda^r(T)$$ and $$\Lambda^r(\operatorname{id_{U}}) = \operatorname{id}|_{\Lambda^r(U)}$$).