My question is on Tensor product of vector spaces and modules.
$(i)$ When we multiply a vector space $V$ with $W$ over a ring (field) $R$ say, then we write $V \otimes_R W$ for the new vector space? Does it mean that we are finding a basis of $V$ and keeping the same basis we are constructing a new vector space over $W$ ? Can you explain $V \otimes_R W$ ?
$(ii)$ Similarly if we consider a free modules $M$ and $N$ over the ring $R$, then $M \otimes_R N$ means to find a basis of $M$ and then keeping teh same basis we are making a new module? Is it true? what is the explanation?
For example, suppose we have free module $M$ over $\mathbb{Q}$, then tensor product by $\mathbb{R}$ i.e., $M \otimes_{\mathbb{Q}} \mathbb{R}$ means we are finding a basis of $M$ and then with the same basis we are constructing a vector space or free module over $\mathbb{R}$. Is it true what I am saying?
Please help me with explanation, specially the last paragraph.
About what you have written I have three steps in my mind:
$\textit{Step 1: }$ First construct the tensor product of two $R-$modules, like $M$ and $N$, namely $M\otimes_R N$. One should consider that $R$ is a ring in this definition.
$\textit{Step 2: }$ We say a module is free when it has a basis. Now let us assume $M$ and $N$ in the previous step be two free modules.
$\textit{Step 3: }$ It is enough to assume that $R$ is not a ring but it is a " field", denote by $F$. So we have two free modules $M$ and $N$over the field $F$. In this case, we have a tensor product of two vector spaces. Because vector spaces are a special case of free modules. (A vector space is a free module over a field)
The point is there are some "facts" which are true about vector spaces but they are not true about free modules. For example, the cardinal of every two bases of a vector space is equal while it is not true for free modules.
$\textit{Edit: }$ For the basis of the resulting vector space, namely $V\otimes_F W$, we would expect them to be built from the basis of $V$ and the basis of $W$. If we have $n$ bases {$ { v_{1}, v_{2}, ... v_{i} }$} for $V$ and {${ w_{1}, w_{2}, ... w_{j} }$} bases for $W$, then we can get a new set of $mn$ vectors. The basis of the resulting vector space will be { $v_{1}\otimes w_{1}$ , $v_{1}\otimes w_{2}$ ,.., $ v_{1}\otimes w_{j}$ ,...., $ v_{2}\otimes w_{1}$, $v_{2}\otimes w_{2}$ ,.., $v_{2}\otimes w_{j}$ ,...., $v_{i}\otimes w_{1}$, $v_{i}\otimes w_{2}$, ...., $v_{i}\otimes w_{j}$ }.
An example of the tensor product of two vectors: Let $V=\mathbb R^3$ and $W=\mathbb R^2$. Take $v_{1}=\begin{bmatrix} 1 \\ 2 \\ 3 \\ \end{bmatrix} $
and $ w_{1}=\begin{bmatrix} 4 \\ 5 \end{bmatrix} $ , then we have $v_{1}\otimes w_{1}= \begin{bmatrix} 1.4 \\ 1.5 \\ 2.4 \\ 2.5 \\ 3.4 \\ 3.5 \\ \end{bmatrix} $ = $\begin{bmatrix} 4 \\ 5 \\ 8 \\ 10 \\ 12 \\ 15 \\ \end{bmatrix} $.