Kernel of tensor product

  • Thread starter ihggin
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  • #1
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Suppose [tex]f_1[/tex] is a linear map between vector spaces [tex]V_1[/tex] and [tex]U_1[/tex], and [tex]f_2[/tex] is a linear map between vector spaces [tex]V_2[/tex] and [tex]U_2[/tex] (all vector spaces over [tex]F[/tex]). Then [tex]f_1 \otimes f_2[/tex] is a linear transformation from [tex]V_1 \otimes_F V_2[/tex] to [tex]U_1 \otimes_F U_2[/tex]. Is there any "nice" way that we can write the kernel of [tex]f_1 \otimes f_2[/tex] in terms of the kernels of [tex]f_1[/tex] and [tex]f_2[/tex]? For example, is it true that [tex]f_1[/tex] and [tex]f_2[/tex] injective implies [tex]f_1 \otimes f_2[/tex] is injective?

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

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  • #2
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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

[tex](f\otimes g)(A)=\langle v,Aw\rangle[/tex]

What can you deduce in this simple case?

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

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