short exact sequence

A sequence of linear maps is exact at a spot if the image of the incoming map equals the kernel of the outgoing one — what enters and dies are the same. A short exact sequence packages the most basic case: 0 -> U -> V -> W -> 0, exact at every interior spot. The two zeros at the ends are the trivial space, and they force the first interior map to be injective and the last to be surjective.

Decode the three conditions. Exactness at U: the image of the map 0 -> U is just {0}, so the kernel of U -> V is {0}, meaning U -> V is injective — U sits inside V as a subspace. Exactness at W: the map W -> 0 has kernel all of W, which must equal the image of V -> W, so V -> W is surjective. Exactness at V: the image of U -> V equals the kernel of V -> W. Putting it together, W is forced to be (isomorphic to) the quotient V/U — this is the first isomorphism theorem dressed as a sequence.

So the canonical short exact sequence is 0 -> U -> V -> V/U -> 0: include the subspace, then quotient it out. Every short exact sequence is, up to isomorphism, of this shape. It is the structural sentence that says V is built from a sub-object U and a quotient-object V/U, no more and no less.

Whether V is actually the DIRECT SUM of those two pieces is a separate question called splitting. A short exact sequence splits if there is a section W -> V putting W back inside V as a complement to U; then V is isomorphic to U (+) W. Over a field every short exact sequence of vector spaces splits — you can always choose a complementary subspace — which is special to vector spaces and fails for general modules. The dimensions always add, dim V = dim U + dim W, exact regardless of splitting.