Subspace of a Free Vector Space spanned by "relations" I am working through some lecture notes which cover an introduction to tensors and the tensor product between vector spaces.
The setup is that $K$ is a field, $S$ is a set and $KS = \oplus_{s \in S}K$ denotes the free vector space built upon the basis $S$.  We let $R$ denote the subspace of $KS$ spanned by all "relations" of the following three types:
$$
(v_1 + v_2) \otimes w - (v_1 \otimes w + v_2 \otimes w)
$$
$$
v \otimes (w_1 + w_2) - (v \otimes w_1 + v\otimes w_2)
$$
$$
v \otimes (\lambda \cdot w) - (\lambda \cdot v) \otimes w - \lambda \cdot (v \otimes w).
$$
The tensor product is defined then by the quotient space $V \otimes W: = KS/R$.  What I don't understand is how you can use "relations" to span a subspace of the free vector space $KS$.  What does the subspace $R$ really look like?  In my head I imagine three different types of subspaces which correspond to all of the relations given above but I know this can't be the correct reasoning, since $R$ will only be one subspace.
Maybe my question is more general to application to constructing a tensor product; how can subspaces be defined using equivalence relations?
 A: First of all, the relations are not correct. They should instead be the following:
$$(v_1+v_2)\otimes w - (v_1 \otimes w + v_1\otimes w),$$
$$v \otimes (w_1 + w_2) - (v \otimes w_1 + v\otimes w_2),$$
$$v \otimes (\lambda \cdot w) - \lambda \cdot (v \otimes w),$$
$$(\lambda \cdot v) \otimes w - \lambda \cdot (v \otimes w),$$
for all $v,v_1,v_2\in V$, all $w,w_1,w_2\in W$ and all $\lambda\in K$, and where $v\otimes w:=(v,w)\in KS$ with $S:=V\times W$.
As for your actual question; the elements of the vector space $KS$, where $S=V\times W$, are precisely the finite sums of the form
$$\sum_{(v,w)\in V\times W}\lambda_{(v,w)}(v,w).$$
That is, they are simply formal finite $K$-linear combinations of pairs $(v,w)\in V\times W$. With the convention that $v\otimes w:=(v,w)\in KS$ we see that the relations above are elements of this vector space $KS$. These elements span some linear subspace $R$, and so we can take the quotient $KS/R$.
For a more tangible example that is similar but a bit contrived, consider the vector space $W=K[X]$ and the subspace $R$ spanned by the functions
$$\lambda X+\mu,\qquad \lambda X^2+\mu,$$
for all $\lambda\in K$. Together these span a subspace $R$ of $W$ that you can describe very explicitly. You can also describe the quotient $W/R$ very explicitly.
The vector space $KS$ with $S=V\times W$ is of course much larger and more abstract. I would recommend not to try to imagine what $R$ looks like in this case. In fact I would recommend to not give this whole construction much thought. As I see it, the construction is an existence proof for $V\otimes W$, where $V\otimes W$ is a vector space defined by a universal property. This property is by far the most relevant property of the tensor product, and is for most purposes the only thing you need to know about a tensor product.
