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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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    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.2011-10-28