understanding the meaning of formal linear combination and tensor product I have a question about understanding the meaning of formal linear combination. 
Let S be a set, the free vector space $\mathbb{R}\langle s\rangle$ on S is defined as the set of all formal linear combination of elements of S with real coefficients. The formal linear combination is a function F: S$\to$$\mathbb{R}$ such that F(s)=0 for all but finitely many s$\in$S. Identifying every element x$\in$S with the function that takes the value 1 on x and zero on other elements of S, any element F$\in$$\mathbb{R}\langle s\rangle$ can be written uniquely in the form F=$\sum_{i=1}^{m}a_ix_i$, where $x_i$ are the elements of S for which F($x_i$)$\neq$0 and $a_i$=F($x_i$).
Here is first question: why we define things like this? why we just say the finite sum of elements in S?
My second question is: Since the free vector space can be used to define tensor product of real vector spaces V and W. Then why we use $\mathbb{R}\langle V\times W\rangle$ modding out some equivalence class instead of use the finite sum of elements in the form $v_i\times w_i$ where $v_i\in V$ and $w_i\in W$ modding out that equivalence class? Thanks for any hint!

Sum of things in $S$? $S$ is just a set; what can the sum of things in $S$ mean?
Once you've constructed $\mathbf{R}\langle s \rangle$, it makes sense to talk about the sum of finitely many things in $S$ using the $+$ operation of $\mathbf{R}\langle s \rangle$ for addition. But without having chosen a $+$ operation, talking about sums is nonsensical.
(also, it's important that we want linear combinations, not merely finite sums)
Regarding the tensor product, we want it to be bilinear. In $\mathbf{R}\langle V \times W \rangle$, it is never true that $(a+b) \times c$ is equal to $a \times c + b \times c$. In fact, $0 \times 0$ is not even the zero vector of $\mathbf{R}\langle V \times W \rangle$!
