(i) Let $V$ be a vector space over a field $F$, and $W_{1}, W_{2}$ subspaces of $V$. Define the subset $W_{1}+W_{2}$ of $V$, and show that $W_{1}+W_{2}$ and $W_{1} \cap W_{2}$ are subspaces of $V$.

(ii) When $W_{1}, W_{2}$ are finite-dimensional, state a formula for $\operatorname{dim}\left(W_{1}+W_{2}\right)$ in terms of $\operatorname{dim} W_{1}, \operatorname{dim} W_{2}$ and $\operatorname{dim}\left(W_{1} \cap W_{2}\right)$.

(iii) Let $V$ be the $\mathbb{R}$-vector space of all $n \times n$ matrices over $\mathbb{R}$. Let $S$ be the subspace of all symmetric matrices and $T$ the subspace of all upper triangular matrices (the matrices $\left(a_{i j}\right)$ such that $a_{i j}=0$ whenever $\left.i>j\right)$. Find $\operatorname{dim} S, \operatorname{dim} T, \operatorname{dim}(S \cap T)$ and $\operatorname{dim}(S+T)$. Briefly justify your answer.