# November 2018

# The Tensor Product, Demystified

Previously on the blog, we've discussed a recurring theme throughout mathematics: making new things from old things. Mathematicians do this all the time:

- When you have two integers, you can find their greatest common divisor or least common multiple.
- When you have some sets, you can form their Cartesian product or their union.
- When you have two groups, you can construct their direct sum or their free product.
- When you have a topological space, you can look for a subspace or a quotient space.
- When you have some vector spaces, you can ask for their direct sum or their intersection.
- The list goes on!

Today, I'd like to focus on a particular way to build a new vector space from old vector spaces: *the tensor product*. This construction often come across as scary and mysterious, but I hope to shine a little light and dispel a little fear. In particular, we won't talk about axioms, universal properties, or commuting diagrams. Instead, we'll take an elementary, concrete look:

Given two vectors $\mathbf{v}$ and $\mathbf{w}$, we can build a new vector, called the *tensor product* $\mathbf{v}\otimes \mathbf{w}$. But what is that vector, *really*? Likewise, given two vector spaces $V$ and $W$, we can build a new vector space, also called their *tensor product* $V\otimes W$. But what is that vector space, *really*?