There's a statement in some notes I'm reading that goes like this: "…$V/U$ is a 'simplified version' of $V$ where the elements of $U$ are ignored" ($V$ and $U$ are vector spaces).
I'm still trying to understand this idea: can someone shed some light on why we ignore $U$ in $V/U$? I mean I understand that everything belonging to $U$ falls in the equivalence class $[0]$, but is that the sense in which we are "ignoring" $U$? Is a quotient space a set of equivalence classes? That is to say that, if we have equivalent affine subsets $v_1 + U = v_2 + U$, are $[v_1]$ and $[v_2]$ the same elements in $U/V$? If this is the case, then aren't we ignoring certain elements of $V$ (like we would ignore $[v_2]$ in this case) because all equivalent affine subsets are redundancies and are therefore collapsed into one respective equivalence class?
Best Answer
In the quotient all elements of $U$ are zero. So in essence we are looking at things outside $U$. Two elements of $V$ are the same in $V/U$ when they differ by an element in $U$. Not only are we looking at things outside $U$, but if they differ by an element of $U$ we consider them the same. So adding elements of $U$ doesn't change anything. It is ignoring $U$ completely, Elements of $U$ are zero and adding by an element of $U$ does nothing.