The definition of $Q/L$ from my course is a bit vague, but I assume it is as follows: if $Q = L + S$ (where “$+$” is the Minkowski sum), then $Q/L = S$.
I realize that if we define $W$ to be the subspace complementary to $L$, then we can write $Q = (Q \cap W) + L$ (i.e. $Q/L = Q \cap W$). But from there I’m still not be able to deduce the dimension of $Q/L$.
Also an extra statement I’d like to show is that $Q/L$ is always pointed, i.e. it always contains a vertex.
Any help would be appreciated.
====================================
Edit: here’s my attempt. Please give some comment, especially if there is some part that need more elaboration.
Let $\text{aff}(Q) = H + v$, where $H$ is a linear subspace (so $H \supseteq L$) of dimension $\text{dim(aff}(Q))$ and $v \neq \textbf{0}$. Let $W$ be a subspace complementary to $L$. Then we have $Q/L = Q \cap W$, and hence $\text{aff}(Q/L) = \text{aff}(Q \cap W) = U + v$, where $U$ is a linear subspace of dimension $\text{dim(aff}(Q \cap W))$ that is also complementary to $L$. Since we have $H = U + L$, and $U \cap L = \textbf{0}$, $H$ is the direct sum of $U$ and $L$, and so: \begin{align*} \text{dim}(H) &= \text{dim}(U) + \text{dim}(L) \\ \Rightarrow \text{dim(aff}(H)) &= \text{dim(aff}(U)) + \text{dim(aff}(L)) \\ \Rightarrow \text{dim}(Q) &= \text{dim}(Q/L) + \text{dim}(L) \end{align*}