sum of subspaces

Given two subspaces U and W of V, their sum U + W is the set of all vectors you can write as u + w with u in U and w in W. Intuitively you pour both subspaces into the same room and let them mix freely by addition. Unlike the union — which is jagged and usually not closed under addition — the sum is automatically a subspace, and in fact the smallest subspace that contains both U and W.

That 'smallest containing' description is the cleaner definition: U + W is the intersection of all subspaces holding U and W, equivalently the span of U together with W. Spelled out, U + W = {u + w : u in U, w in W}. The two descriptions agree, and either way you get a genuine subspace closed under addition and scaling, sitting above both U and W in the subspace lattice as their join.

The headline result is the dimension formula: dim(U + W) = dim U + dim W - dim(U ∩ W). You add the two dimensions, then subtract the overlap you double-counted. If two planes in R^3 meet in a line, the sum has dimension 2 + 2 - 1 = 3 — it fills all of R^3. The formula is the linear-algebra echo of inclusion-exclusion, and it is the gateway to the direct sum, where the overlap is forced to zero.

A subtlety to respect: the representation u + w is generally NOT unique. The same vector in U + W can split into u + w in many ways precisely when U ∩ W is nontrivial — any vector in the overlap can shuffle between the two pieces. Demanding uniqueness of the split is exactly the extra condition that promotes a plain sum to a DIRECT sum.