Let $U,W$ be [[vector space|vector spaces]] over [[field]] $K$ The **direct product**, the set containing [[tuple|tuples]] $(u,w)$ from the spaces $U$ and $W$, is $U\times W$ $U\times W = \set{(u,w):u\in U\ \land \ w\in W}$ Tuples from the product can be added $\left(u_{1}, w_{1}\right)+\left(u_{2}, w_{2}\right)=\left(u_{1}+u_{2}, w_{1}+w_{2}\right)$ And they can be scaled for $c \in K$ $ c\left(u_{1}, w_{1}\right)=\left(c u_{1}, c w_{1}\right) $ Since both spaces contain a zero vector $(\mathbf0,\mathbf0)\in\ U\times W$ Which works as an additive identity. So this direct product works as its own vector space In the case that a **direct product** is taken of a [[set]] of $n$ vector spaces $\begin{aligned}\prod_{i=1}^{n} W_{i}&=W_{1} \times \cdots \times W_{n}\\&=\set{\left(w_{1}, \ldots, w_{n}\right): w_{i} \in W_{i}}\end{aligned}$