## Decomposition of tensor product space [closed]

i have one fundamental question regarding tensor product.

Let $\mathbf{v}_1$, $\mathbf{v}_2$ and $\mathbf{v}_3$ be the basis of the vector space V. Let $\mathbf{w}_1$, $\mathbf{w}_2$ and $\mathbf{w}_3$ be the basis of the vector space W. Let $\mathbf{g}_i$ for i = 1,2,...,9 be the basis of the tensor product space given by V $\otimes$ W.

NOT all the vectors, $\mathbf{q}_i$, in the tensor product space can be written in the form $\mathbf{q}_i = \mathbf{v}_j \otimes \mathbf{w}_k$ but only a subset of vectors.

The question is as follows: Given a tensor product space, how to find the set of vectors $\mathbf{q}_i$ in the tensor product space VxW such that it can be represented as the tensor product of any two vectors.

thank you.

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This would be better asked at math.stackexchange. But the question doesn't make sense-- what does "how can we find" mean? You have just written down $v_j\otimes w_k$ so what sort of criteria are you looking for? It's like you've asked "How can we find the set of numbers $n$ such that $n=5$?" – Matthew Daws Sep 20 2011 at 14:45
i have editted the question to make it more clear. Actually, i mean the tensor product space is given. if i randomly take a vector in this product space it cannot be written as the tensor product of two vectors. but there are some vectors in this product space which can be written as the tensor product of two vectors. i am searching for such vector in the tensor product space. – karthick Sep 20 2011 at 15:04
The edit is slightly better-- but why not just read Wikipedia (the "tensor" article, then search for "rank"). Really-- please ask this on math.stackexchange.com – Matthew Daws Sep 20 2011 at 15:04
ok. Thanks i will read and ask in math.stackexchange.com. – karthick Sep 20 2011 at 15:05