Practical Uses for the Tensor Product From reading a bit of chapter section 18 in Differential Geometry, by Loring W. Tu, my understanding of tensor products is that they allow one to represent multilinear maps with linear maps.
I was simply curious whether there is any practical use for the tensor product (perhaps in computer science or physics, etc.). Are there other interpretations of tensor products?
 A: If $\varepsilon^1,\varepsilon^2$ are the projectors on $\mathbb R^2$, that is for any $v=(v^1,v^2)^{\top}$ then $\varepsilon^i(v)=v^i$, so the tensor product
$$\varepsilon^1\wedge\varepsilon^2=\varepsilon^1\otimes\varepsilon^2-\varepsilon^2\otimes\varepsilon^1,$$
satisfies, for a pair of vectors
$$\varepsilon^1\wedge\varepsilon^2(v,w)=v^1w^2-w^1v^2,$$
which is the determinant of the data given by $v,w$ and at the same time the area of the parallelogram that vector $v,w$ spawn.
A: In quantum mechanics, the state of a physical system (say, a particle) is an element of a Hilbert space $\mathcal H$. If we take two such systems (say, two particles) with Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, and consider them together, then the state of the physical system will be an element of the tensor product $\mathcal H_1\otimes\mathcal H_2$ of Hilbert spaces.
Also, in relativity, we  work on a pseudo-Riemannian manifold $M$, on which we have a metric which gives us the spacetime interval between two events. A metric on a pseudo-Riemannian manifold is just a non-degenerate, symmetric bilinear form on every tangent space $T_pM$ of the manifold, which we can interpret as an element from the tensor product $T_pM^\ast\otimes T_pM^\ast$ of the duals of the tangent spaces.
A: Physics: Good examples are provided on the wikipedia page for tensors.
CS: Tensors are a fundamental part of machine learning. It's not for nothing that Google's machine learning software is called TensorFlow
Math: Tensors are used widely, but one notable area is representation theory
In general, tensors are essentially higher dimensional versions of matrices, so it's not completely surprising that they would arise in certain settings.
