direct sum

A direct sum is a sum of subspaces in which the splitting is unambiguous. We write V = U ⊕ W and call it a direct (internal) sum when V = U + W AND U ∩ W = {0}. The vanishing intersection is the whole point: it forces every vector v in V to have ONE and only one decomposition v = u + w with u in U and w in W. No vector can shuffle between the pieces, so the two subspaces are genuinely independent coordinates for V.

There is a second, external flavor. Given any two spaces U and W — not necessarily living inside a common V — you can build a brand-new space U ⊕ W of ordered pairs (u, w), added and scaled componentwise. This is how R^2 is literally R ⊕ R. The internal and external versions agree up to canonical isomorphism: an internal V = U ⊕ W is isomorphic to the external pair space via v = u + w <-> (u, w). One viewpoint decomposes a space you already have; the other assembles a new one from parts.

Direct sums are the cleanest way to break a hard problem into independent easy ones. If an operator T preserves both U and W in V = U ⊕ W, then T splits into its restrictions to each piece, and its matrix becomes block-diagonal. Eigenspace decomposition, the primary decomposition, and ultimately Jordan and spectral theory are all stories of writing V as a direct sum on which the operator acts one block at a time.

The trivial-intersection condition is more delicate than it looks for three or more summands. V = U_1 ⊕ U_2 ⊕ U_3 is NOT guaranteed just because the spaces pairwise meet in {0}; you need each U_i to meet the sum of the others in {0}, or equivalently that the zero vector has only the all-zero decomposition. Pairwise independence is strictly weaker than joint independence — a trap worth remembering.