# Tensor products and linear transformations

If $E$ and $F$ are linear transformations. How does one prove that $rank(A \otimes B)=rank(A) rank(B)$ where $\otimes$ is the tensor product. This is a question I do not know how to approach. Can I get some help?

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Sorry, I can't quite comment yet. Try to prove (an ostensibly easier thing to do) that if $A:V\to W$ and $B:U\to Z$ (so that $A\otimes B:V\otimes W\to U\otimes Z$ then $\text{im }(A\otimes B)=(\text{im}(A))\otimes(\text{im}(B))$. This should be very clear by definition. and then apply the fact that dimension is multiplicative with respect to tensor products.

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You can comment now, but so what? This is perfectly good as an answer. –  Henning Makholm Oct 28 '11 at 0:42
Thank you Henning, I am new here, and wasn't sure if such a paltry response warranted a "answer"--some of the answers are full-on explanations, not hints. –  Alex Youcis Oct 28 '11 at 0:44
If anything, our problem is that too many answers that turn out to solve the asker's problem and close the case are posted as comments instead. Then the question will still turn up in lists marked as "0 answers" which makes it difficult to recognize places where more work is needed. On the other hand, there is no real harm in having a partial answer be an answer rather than a comment. If a better answer arrives later, the voting system will ensure that it floats to the top quickly, so no harm will result. –  Henning Makholm Oct 28 '11 at 0:56
Thanks for the help. –  smanoos Oct 28 '11 at 3:13