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.