What Can be Said About the Kernel of a Tensor Product of Linear Maps?

[ihggin]

Suppose $$f_1$$ is a linear map between vector spaces $$V_1$$ and $$U_1$$, and $$f_2$$ is a linear map between vector spaces $$V_2$$ and $$U_2$$ (all vector spaces over $$F$$). Then $$f_1 \otimes f_2$$ is a linear transformation from $$V_1 \otimes_F V_2$$ to $$U_1 \otimes_F U_2$$. Is there any "nice" way that we can write the kernel of $$f_1 \otimes f_2$$ in terms of the kernels of $$f_1$$ and $$f_2$$? For example, is it true that $$f_1$$ and $$f_2$$ injective implies $$f_1 \otimes f_2$$ is injective?

I tried assuming $$f_1 \otimes f_2$$ acting on a general element $$\sum v_1 \otimes v_2$$ was zero, but the resulting tensor $$\sum f_1(v_1) \otimes f_2(v_2)$$ is too complicated for me to draw implications for $$v_1$$ and $$v_2$$. It is obvious that $$v_1 \in \ker f_1$$ or $$v_2 \in \ker f_2$$ implies that the latter tensor product is 0, but what can be said for the other direction?

 

[arkajad]

I suggest to look at a simple case, when f and g are linear functionals on the same vector space. Then they are given by a pair of (dual) vectors, say v and w, and an element of the tensor product of vector space by itself is essentially a matrix, say A.
Then you will get something like

$$(f\otimes g)(A)=\langle v,Aw\rangle$$

What can you deduce in this simple case?

I hope my reasoning is roughly correct, but I was making just quick intuitive guesses.