Let $V$ and $W$ be two real vector spaces, $v \in V$ and $w \in W$. I'm getting some trouble in the following problem:

Let $u_1$ and $u_2$ be two elements in the tensor $V \otimes W$ such that $u_1 + u_2 = v \otimes w$. I'm asking myself if this implies that

  • $u_1 = v_1 \otimes w$ and $u_2 = v_2 \otimes w$ with $v_1 + v_2 = v$, or
  • $u_1 = v \otimes w_1$ and $u_2 = v \otimes w_2$ with $w_1 + w_2 = w$.

This seems reasonable to happen, but I'm not so used to work with tensor products. I'd like some reference to study this kind of situation if possible.

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    $\begingroup$ Notice your question is equivalent to "is every tensor a simple tensor?" (i.e. of the form $v \otimes w$) to which the answer is no. Assuming that every tensor is simple is the number 1 mistake made when first working with them. $\endgroup$ – RghtHndSd Jul 1 at 15:00

No you could have $v\otimes w=(v\otimes w -v_1\otimes w_1-v_2\otimes w_2)+(v_1\otimes w_1+v_2\otimes w_2)$. This does not in general have either of the forms you suggested.

Tensor products take almost everyone a while to get their heads around when first encountered. You are asking good questions in order to do this.

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  • $\begingroup$ If you note the two tensors you took and sum $v - v_1 - v_2 + v_1 + v_2 = v$ and $w - w_1 - w_2 + w_1 + w_2$. This is a fenomenum that always happen? $\endgroup$ – user 242964 Jul 1 at 15:33
  • $\begingroup$ No you could have $v\otimes w=(v\otimes w -v_1\otimes 2w_1-v_2\otimes w_2)+(2v_1\otimes w_1+v_2\otimes w_2)$. $\endgroup$ – tkf Jul 1 at 15:46

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