I am a physics student and this question mostly pertains to trying to understand the tensor product. I'm trying to see the relationship between column vector and matrix forms of the tensor product of two vectors.
My question is this, when we take the vector space $\Bbb R^{n\times n}$ and want to show it is isomorphic to the vector space of $n\times n$ real matrices, does it matter which component in the vector is mapped to which component in the matrix?
As an example, say I want to define a relationship between $v\in \Bbb R^4$ and the corresponding $2\times 2$ real matrix under the isomorphism, is the map: $$I:v_i\mapsto M_{jk},\quad i=1,2,3,4,\quad j,k=1,2$$ equivalent for all $i$,$j$ and $k$ if the only thing we are interested in preserving is the linear structure? Assuming it is 1-to-1 and performed the same way for all elements of $V$.
Let $i=nq+r$ where $0\le r\le n-1$, then $v_i\mapsto M_{q+1,r+1}$.