Suppose $U$ and $W$ are subspaces of a vector space $V$. Explain what is meant by $U \cap W$ and $U+W$ and show that both of these are subspaces of $V$.

Show that if $U$ and $W$ are subspaces of a finite dimensional space $V$ then

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

Determine the dimension of the subspace $W$ of $\mathbb{R}^{5}$ spanned by the vectors

$$\left(\begin{array}{c} 1 \\ 3 \\ 3 \\ -1 \\ 1 \end{array}\right),\left(\begin{array}{l} 4 \\ 1 \\ 3 \\ 2 \\ 1 \end{array}\right),\left(\begin{array}{l} 3 \\ 2 \\ 1 \\ 2 \\ 3 \end{array}\right),\left(\begin{array}{c} 2 \\ 2 \\ 5 \\ -1 \\ -1 \end{array}\right)$$

Write down a $5 \times 5$ matrix which defines a linear map $\mathbb{R}^{5} \rightarrow \mathbb{R}^{5}$ with $(1,1,1,1,1)^{T}$ in the kernel and with image $W$.

What is the dimension of the space spanned by all linear maps $\mathbb{R}^{5} \rightarrow \mathbb{R}^{5}$

(i) with $(1,1,1,1,1)^{T}$ in the kernel and with image contained in $W$,

(ii) with $(1,1,1,1,1)^{T}$ in the kernel or with image contained in $W$ ?