Define what is meant by a vector space $V$ over the real numbers $\mathbb{R}$. Define subspace, proper subspace, spanning set, basis, and dimension.

Define the sum $U+W$ and intersection $U \cap W$ of two subspaces $U$ and $W$ of a vector space $V$. Why is the intersection never empty?

Let $V=\mathbb{R}^{4}$ and let $U=\left\{\mathbf{x} \in V: x_{1}-x_{2}+x_{3}-x_{4}=0\right\}$, where $\mathbf{x}=\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$, and let $W=\left\{\mathbf{x} \in V: x_{1}-x_{2}-x_{3}+x_{4}=0\right\}$. Show that $U \cap W$ has the orthogonal basis $\mathbf{b}_{1}, \mathbf{b}_{2}$ where $\mathbf{b}_{1}=(1,1,0,0)$ and $\mathbf{b}_{2}=(0,0,1,1)$. Extend this basis to find orthogonal bases of $U, W$, and $U+W$. Show that $U+W=V$ and hence verify that, in this case,

$$\operatorname{dim} U+\operatorname{dim} W=\operatorname{dim}(U+W)+\operatorname{dim}(U \cap W)$$