Specific example of the universal property of $V \otimes W$ I'm trying to familiarize myself with the definition of tensors, so I was wondering if I understood the definition in terms of the universal property.
Consider a bilinear $B : V \times W \rightarrow U$. That is, $B(\sum \alpha_i v_i, \sum \beta_j w_j) = \sum_{i,j} \alpha_i \beta_j B(v_i,w_j) \in U$.
Then I can quotient out the desired equivalence relation to get $T : V \times W \rightarrow V \otimes W$, which is of the form $T(\sum \alpha_i v_i, \sum \beta_j w_j) = \sum_{i,j} \alpha_i \beta_j (v_i \otimes w_j)$. Then the unique linear map $\ell$ is of the form:
$$\sum \alpha_i \beta_j (v_i \otimes w_j) \mapsto \sum_{k=1}^{ij} \gamma_k \ell((v \otimes w)_k)$$
where $k$ ranges over every ordered pair $(i,j)$, and is $\gamma_k = \alpha_i\beta_j$, and similarly for each basis of $V \otimes W$.
Is this correct? Or am I way off?
 A: I think you have the right idea, but you're overcomplicating it a bit.
By the universal property of $V\otimes W$, there is a unique linear map $\ell : V\otimes W \to U$ such that $B = \ell\circ T$. For $(v, w) \in V\times W$ we have $T(v, w) = v\otimes w$, so
$$\ell(v\otimes w) = \ell(T(v, w)) = (\ell\circ T)(v, w) = B(v, w).$$
Note that a generic element of $V\otimes W$ is not of the form $v\otimes w$, but it is a sum of such elements. For $\sum_{i=1}^nv_i\otimes w_i \in V\otimes W$, we have $$\ell\left(\sum_{i=1}^nv_i\otimes w_i\right) = \sum_{i=1}^nB(v_i, w_i)$$
as $\ell$ is linear.
A: Here is the example that really made it "click" for me:
Let $V = W = \mathbb{R}^n$.
Let the bilinear map $B: V\times W = \mathbb{R}^n  \times \mathbb{R}^n \to U = \mathbb{R}$ be the standard dot product.
Identify $V \otimes W$ with $\mathbb{R}^{n \times n}$, the space of $n \times n$ real-valued matrices.
Then the corresponding unique linear map $\ell: V \otimes W = \mathbb{R}^{n  \times n} \to \mathbb{R} = U$ is the trace.
In particular, show that for any two vectors $v, w$, their dot product $v \bullet w$ equals the trace $tr (v \otimes w)$ of the rank-1 matrix $v \otimes w$ defined by the so-called "outer product", i.e. $T: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^{n \times n}$ with $T: (v,w) \mapsto v \otimes w$.
Anyway, if you can walk yourself through this example, which is easy to understand and to calculate, and convince yourself that it fits into the general definition, the general definition will be easier to understand (at least it was for me).
(As an aside, I think "outer product" is not great terminology, because it is easy to confuse with the "exterior product", which is completely different, and the terminology suggests a contrast/relationship with general "inner products", when in fact it is really only related to the particular inner product that is the dot product.)
