We mentioned this in class, and I figured we should write it down.

So, if you think about the Product of Vector Space:

\begin{equation} \mathbb{R} \times \mathbb{R} \end{equation}

you are essentially taking the \(x\) axis straight line and “duplicating” it along the \(y\) axis.

Now, the opposite of this is the quotient space:

\begin{equation} \mathbb{R}^{2} / \left\{\mqty(a \\ 0): a \in \mathbb{R} \right\} \end{equation}

Where, we are essentially taking the line in the \(x\) axis and squish it down, leaving us only the \(y\) component freedom to play with (as each element is \(v +\left\{\mqty(a \\ 0): a \in \mathbb{R} \right\}\)).

This also gets us the result that two affine subsets parallel to \(U\) are either equal or disjoint; specifically the conclusion that \(v-w \in U \implies v+U = w+U\): for our example, only shifting up and down should do different things; if two shifts’ up-down shift is \(0\) (i.e. it drops us back into \(\mqty(a \\0)\) land), well then it will not move us anywhere different.