quotient space
The quotient space V/U is what you get when you decide to stop distinguishing vectors that differ by something in U. Choose a subspace U to 'ignore'; then declare two vectors equal whenever their difference lies in U. The resulting clumps of identified vectors are the elements of V/U. Loosely, you collapse all of U down to a single point — the new zero — and V/U records only the information that survives that collapse.
Each element of V/U is a coset v + U = {v + u : u in U}, the set of all vectors reachable from v by sliding within U. Addition and scaling are defined on cosets in the obvious way — (v + U) + (w + U) = (v + w) + U and a*(v + U) = (a*v) + U — and the miracle is that these are well-defined: the answer does not depend on which representative v you grabbed from the coset. With these operations V/U is itself a genuine vector space.
The dimension bookkeeping is clean and memorable: dim(V/U) = dim V - dim U. Collapsing a d-dimensional subspace removes exactly d dimensions. If V is R^3 and U is a plane through the origin, V/U is one-dimensional — it records only 'which parallel plane am I on', having thrown away all motion within each plane. The quotient keeps the directions transverse to U and forgets the rest.
Why quotient instead of just picking a complement? Because the quotient is canonical — it needs no choice — whereas a complement is one of infinitely many. V/U is isomorphic to any complement of U, but it does not privilege one, which makes it the right tool for clean theorems. The first isomorphism theorem, V/ker(T) isomorphic to im(T), is the quotient earning its keep: it turns rank-nullity into a structural statement rather than a dimension count.
Collapsing U removes dim U dimensions; coset arithmetic is just representative arithmetic, and it is well-defined.
The quotient map q: V -> V/U sending v to its coset v + U is linear, surjective, and has kernel exactly U. So every subspace is the kernel of some linear map — namely its own quotient map — which is the converse of the fact that every kernel is a subspace.