Abstract Linear Algebra 39 | Direct Sum of Subspaces

Direct sums of subspaces are introduced as the key tool for splitting a vector space into two parts that don’t overlap except at the zero vector—and that non-overlap is exactly what makes the decomposition useful for later Jordan normal form work. Starting with two subspaces U1 and U2 inside a vector space V, simply taking their union usually fails to produce a subspace because linear combinations of vectors from the union may leave the set. The fix is to form U1 + U2, meaning all sums u1 + u2 with u1 ∈ U1 and u2 ∈ U2; this always yields a subspace.

The direct sum tightens this idea by adding a crucial condition: U1 ⊕ U2 is defined only when U1 ∩ U2 = {0}. In geometric terms, the subspaces point in completely different directions, so no nonzero vector can belong to both. Under this condition, every vector in U1 ⊕ U2 has a clean representation as a sum of one vector from each subspace, and the dimension behaves additively: dim(U1 ⊕ U2) = dim(U1) + dim(U2). A simple example in C^2 illustrates the point: taking span{(1,0)} and span{(0,1)} gives C^2 as a direct sum of two one-dimensional subspaces.

With direct sums in place, the discussion pivots to Jordan normal form. Fix a complex vector space C^n and a square matrix A. For an eigenvalue λ, define N = A − λI. Prior material established chains of generalized eigenspaces through kernels of powers of N, K(N^0) ⊆ K(N^1) ⊆ …, which stabilize at the fitting index D (so K(N^D) = K(N^{D+1}) and similarly for larger powers). The same fitting index also appears in the corresponding chain of ranges: R(N^0) ⊇ R(N^1) ⊇ …, with stabilization at D.

At the fitting index level, the space C^n can be decomposed as a direct sum of the kernel and the range: C^n = K(N^D) ⊕ R(N^D). The argument for “direct” hinges on proving the intersection is trivial. If X lies in K(N^D) ∩ R(N^D), then X = N^D u for some u (because X is in the range) and also N^D X = 0 (because X is in the kernel). Substituting X = N^D u gives N^D(N^D u) = N^{2D}u = 0, which forces u into K(N^D) once stabilization at the fitting index is used. Then X = N^D u becomes 0, so the intersection contains only the zero vector.

Finally, the decomposition is shown to cover all of C^n. Since the direct sum has dimension equal to dim(K(N^D)) + dim(R(N^D)), the rank-nullity theorem supplies the total as n. An n-dimensional subspace of C^n must be the whole space, so K(N^D) ⊕ R(N^D) = C^n. This kernel–range splitting for each eigenvalue is presented as the structural ingredient that feeds directly into the Jordan normal form construction in the next step.