How to prove that $f\otimes g: V\otimes W\to X\otimes Y$ is a monomorphism Let $V, W, X, Y$ be the vector spaces over a field $\mathbb K$. Suppose that $f: V\to X$ and $g: W\to Y$ are the homomorphism of $\mathbb K$-vector spaces.
How to prove that $f\otimes_{\mathbb K} g: V\otimes_{\mathbb K} W\to X\otimes_{\mathbb K} Y$ is a monomorphism map if and only if $f$ and $g$ are also monomorphism maps?
Thanks all for help!

A basic structure theorem for vector spaces is that given any linear map
$$ T : A \to B $$
you can decompose everything into direct sums:
$$ A \cong \ker(T) \oplus \operatorname{coim}(T) \qquad \qquad B \cong \operatorname{im}(T) \oplus \operatorname{coker}(T) $$
such that $T$ restricts to the canonical isomorphism $\operatorname{coim}(T) \to \operatorname{im}(T)$.

You can use the fact tensor products distribute over direct sums to decompose your problem into smaller pieces that are easy to analyze.
