Let $V$ be a vector space of dimension 3 with the basis {$e_1,e_2,e_3$} and let $W$ be a vector space of dimension 2 with basis {$f_1,f_2$}. Which of the following tensors are decomposable (i.e. of the form $v\otimes w$ with $v\in V, w\in W$).

a) $s=e_1\otimes f_1+e_2\otimes f_2$,

b) $t=e_1\otimes f_2+e_1\otimes f_3-e_2\otimes f_2-e_2\otimes f_3$

c) $u=e_1\otimes f_2-e_1\otimes f_3+2e_2\otimes f_2$.

So, what is the condition for a tensor $$v=\sum_{i=1}^3\sum_{j=1}^2a_{i,j}e_i\otimes f_j$$ from $V\otimes W$ to be decomposable? Is there any existing testing methods(for example, some function of $a_{i,j}$)? or is there any specific properties of tensor that is crucial here to find $v\in V$ and $w\in W$?

Thank you very much!


The vector $v$ (as you've expressed it) will be decomposable if and only if the matrix with entries $a_{ij}$ has rank at most $1$.

To see why, compare the product $uv^T$ to the coefficients of $u\otimes v$.


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