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Let $k$ be a a field and consider the space $k[x] \otimes_k k[x]$. I would like to verify the equation

$ \sum_{k=0}^{m+n} {m+n \choose k} x^k \otimes x^{(n+m)-k}= \sum_{i=0}^n \sum_{j=0}^m{n \choose i}{m \choose j} x^{i+j}\otimes x^{(n+m)-(i+j)}$

however I am not getting very far with this. Could anyone give me some help? Thanks very much.

1 Answers 1

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This doesn't really have much to do with tensor products. If you group together the terms on the right hand side with a common value of $i+j$ (call the common value $k$), then the identity reduces to the binomial coefficient identity $\sum_{i+j=k} \binom{n}{i} \binom{m}{j} = \binom{n+m}{k}.$ This identity is proved by a combinatorial argument or by comparing the two sides of $(1+x)^{m+n} = (1+x)^m (1+x)^n$.