# Tensor Product The **tensor product** is a way to build a new vector space from old vector spaces. We can start with a simple example. Given two vectors, $\bf{v}$ and $\bf{w}$, we can build a new vector called the *tensor product*: $\bf{v} \otimes \bf{w}$ But what is that vector, *really*? Likewise, if we have two [Vector Spaces](Abstract%20Vector%20Spaces.md), $V$ and $W$, we can build a new vector space, also called their *tensor product*: $V \otimes W$ But what is that vector space, *really*?    --- Date: 20230727 Links to: Tags: References: * [The Tensor Product, Demystified](https://www.math3ma.com/blog/the-tensor-product-demystified)