Tensor Product Vector Space Example While trying to study representation theory, I came across the tensor product of vector spaces - a new topic to me (although done some multinear algebra in the past). A definition I came across of the tensor product of vector spaces $U$ and $V$ is that it is the vector space $U\otimes{V}$ such that bilinear maps from $U\times{V}$ into a vector space  W, are linear maps from $U\otimes{V}$ onto W. In this way, $U\otimes{V}$ reduces multilinear algebra to the more simple linear algebra.
I am trying currently to now find an example in understanding this topic. I considered letting $U=\mathbb{R^3}$, $V=\mathbb{R^3}$, $W=\mathbb{R^3}$, and then considering the bilinear map that is the cross product from $U\times{V}$ into W. However, I can't seem to construct the space $U\otimes{V}$ such that the cross product is a linear map from $U\otimes{V}$ into $W$. Could someone help me with identifying this space/supplement with further examples of the tensor product of vector spaces. Thanks.
 A: Let us consider the basis of $U \otimes V$. As we are dealing with $\mathbb{R}^3 \otimes \mathbb{R}^3$, the basis $B$ is the set of all $e_i \otimes e_j$ for $i,j \in \{1, 2, 3\}$. Note that for the cross product, the multiplication table is as follows
$$
\begin{array}{c|c|c|c|}
\times & e_1 & e_2 & e_3 \\
\hline
e_1 & 0 & e_3& -e_2\\
\hline 
e_2 & -e_3 & 0 & e_1 \\
\hline 
e_3 & e_2 & -e_1 & 0 \\
\hline 
\end{array}
$$
Then define a map $W: B \to \mathbb{R}^3$ by the following multiplication table
$$
\begin{array}{c|c|c|c|}
W & e_1 & e_2 & e_3 \\
\hline
e_1 & 0 & e_3& -e_2\\
\hline 
e_2 & -e_3 & 0 & e_1 \\
\hline 
e_3 & e_2 & -e_1 & 0 \\
\hline 
\end{array}
$$
As an example of how the table is to be read, this table says that $W(e_1 \otimes e_2) = e_3$.
Now that we have defined $W$ on the basis, we can extend to the rest of $\mathbb{R}^3 \otimes \mathbb{R}^3$. This is because the tensor product and the cross product have the same structure. The cross product is a bilinear map, that is:
$$
\begin{align}
(v_1 + v_2) \times w = v_1 \times w + v_2 \times w \\
v \times (w_1 + w_2) = v \times w_1 + v \times w_2 \\
\end{align}
$$
and the tensor space follows a set of "bilinear" axioms:
$$
\begin{align}
(v_1 + v_2) \otimes w = v_1 \otimes w + v_2 \otimes w \\
v \otimes (w_1 + w_2) = v \otimes w_1 + v \otimes w_2 \\
\end{align}
$$
These are very similar. Now, extending $W$ linearly, we truly see that 
$$W(v \otimes w) = v \times w$$
As an example, which illuminates the proof of this fact, consider the following computation:
$$
\begin{align}
W((1\mathbf{e}_1 + 2\mathbf{e}_3) \otimes (3\mathbf{e}_2 + 4\mathbf{e}_3)) &= \\
W(1\mathbf{e}_1 \otimes (3\mathbf{e}_2 + 4\mathbf{e}_3) + 2\mathbf{e}_3 \otimes (3\mathbf{e}_2 + 4\mathbf{e}_3)) &= \\
W(1\mathbf{e}_1 \otimes 3\mathbf{e}_2 + 1\mathbf{e}_1 \otimes 4\mathbf{e}_3 + 2\mathbf{e}_3 \otimes 3\mathbf{e}_2 + 2\mathbf{e}_3 \otimes 4\mathbf{e}_3) &= \\
W(1\mathbf{e}_1 \otimes 3\mathbf{e}_2) + W(1\mathbf{e}_1 \otimes 4\mathbf{e}_3) + W(2\mathbf{e}_3 \otimes 3\mathbf{e}_2) + W(2\mathbf{e}_3 \otimes 4\mathbf{e}_3) &= \\
W(3(\mathbf{e}_1 \otimes \mathbf{e}_2)) + W(4(\mathbf{e}_1 \otimes \mathbf{e}_3)) + W(6(\mathbf{e}_3 \otimes \mathbf{e}_2)) + W(8(\mathbf{e}_3 \otimes \mathbf{e}_3)) &= \\
3W(\mathbf{e}_1 \otimes \mathbf{e}_2) + 4W(\mathbf{e}_1 \otimes \mathbf{e}_3) + 6W(\mathbf{e}_3 \otimes \mathbf{e}_2) + 8W(\mathbf{e}_3 \otimes \mathbf{e}_3) &= \\
3\mathbf{e}_3 - 4\mathbf{e}_2 - 6\mathbf{e}_1 + 8\cdot\mathbf{0} &= \\
-6\mathbf{e}_1 - 4\mathbf{e}_2 + 3\mathbf{e}_3
\end{align}
$$
which is indeed the cross product of those two vectors. Steps 4 and 6 rely on the fact $W$ had been extended linearly, while the rest of the steps used the bilinear properties of the tensor product. (Compare this to explicitly finding the cross product by expanding and multiplying!)

Conclusion: The tensor product allows us to use its axioms of bilinearity to "mimic" the properties of a bilinear object or map.
A: Let $U$ be the space of real polynomials in the "formal variable" $x$ with degree $\leq 3$. $U$ is an $\mathbb{R}$-vector space, as you can easily verify. What does the canonical basis of $U$ look like? I bet you already know that it is
$$B_U =\{x^0, x^1, x^2, x^3\}$$
Here is an example element of $U$: 
$$3x^3 + x^2 + 10$$
Let's also consider the vector space $V$ of real polynomials in the "formal variable" $y$ with degree $\leq 2$. This time, the basis is 
$$B_V = \{y^0, y^1, y^2\}$$
Here is an example element of $V$:
$$11y^2 - 9$$
Note that we are only consider the "form" of the variable, in this sense, because $x^0$ and $y^0$ are different objects, they are not considered the same.

Now by definition, here is the basis of $B_U \otimes B_V$.
$$B_{U \otimes V} = 
\begin{Bmatrix}
x^0 \otimes y^0 & x^1 \otimes y^0 & x^2 \otimes y^0 & x^3 \otimes y^0 \\
x^0 \otimes y^1 & x^1 \otimes y^1 & x^2 \otimes y^1 & x^3 \otimes y^1 \\
x^0 \otimes y^2 & x^1 \otimes y^2 & x^2 \otimes y^2 & x^3 \otimes y^2 \\
\end{Bmatrix}
$$
Using this basis, here is an element of $U \otimes V$.
$$5(x^3 \otimes y^2) + 4(x^3 \otimes y^1) - 10(x \otimes y^2) + 6(x^0 \otimes y^1) - 10(x^0 \otimes y^0)$$
See how this looks like a polynomial in $x$ and $y$? The tensor product space acts exactly in the way you'd think, using bilinearity(acting as distributivity) to allow us to "multiply" the elements of two vector spaces. Using the examples from before,
$$(3x^3 + x^2 + 10) \otimes (11y^2 - 9) = \\ 
33(x^3 \otimes y^2) + 11(x^2 \otimes y^2) + 110(x^0 \otimes y^2) - 27(x^3 \otimes y^0) - 9(x^2 \otimes y^0) - 90(x^0 \otimes y^0)$$
Which looks like the actual product
$$(3x^3 + x^2 + 10)(11y^2 - 9) = \\ 
33x^3y^2 + 11x^2y^2 + 110y^2 - 27x^3 - 9x^2 - 90$$

Conclusion: The tensor product space is a space where the elements from each original space may be multiplied. This multiplication is distributive and allows different basis elements to interact, meaning it looks a lot like polynomial multiplication.
Note: Not every element in $U \otimes V$ is a multiplication of elements in $U$ and $V$. The example element of $U \otimes V$ I gave before is not factorable, as an example. Look here for more information. 
