JOVANA
Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

Direct Sums and Complements: Splitting a Space

When two subspaces overlap only at 0, their sum becomes a [[direct-sum|direct sum]] V = U ⊕ W: every vector splits *uniquely*. We meet the [[complementary-subspace|complementary subspace]] and learn to decompose a space cleanly.

The uniqueness that defines a direct sum

A sum U + W is called a [[direct-sum|direct sum]], written U ⊕ W, when the decomposition is unique: every v in U + W can be written as u + w with u in U and w in W in exactly one way. The single algebraic condition that guarantees this is U ∩ W = {0}.

Why U ^ W = {0} forces uniqueness:
   suppose  v = u1 + w1 = u2 + w2
   then     u1 - u2 = w2 - w1
   left side in U, right side in W, so the common value
   lies in U ^ W = {0}.
   => u1 - u2 = 0 and w2 - w1 = 0  =>  u1=u2, w1=w2.  Unique!
Trivial intersection is exactly equivalent to unique decomposition.

Complements: every subspace has a partner

Given a subspace U of a finite-dimensional V, a [[complementary-subspace|complementary subspace]] W is any subspace with V = U ⊕ W. Complements always exist — extend a basis of U to a basis of V, and the new vectors span a W that works. Note: complements are not unique. In R^2, a line U has *many* complementary lines.

Worked split of M_2 and many summands

Split M_2 (2x2 matrices) into symmetric + skew-symmetric:
   S = { A : A^T = A }   dim 3   (basis [1,0;0,0],[0,0;0,1],[0,1;1,0])
   K = { A : A^T = -A }  dim 1   (basis [0,1;-1,0])

Every matrix splits uniquely:
   A = (A + A^T)/2  +  (A - A^T)/2
        ^symmetric        ^skew
   S ^ K = {0}  and  dim S + dim K = 3 + 1 = 4 = dim M_2
   =>  M_2 = S (+) K.
M_2 = symmetric ⊕ skew-symmetric — a canonical direct-sum decomposition.

Direct sums chain: V = U_1 ⊕ U_2 ⊕ ... ⊕ U_k means every vector splits uniquely across all k pieces. This is the structural engine behind diagonalization — a diagonalizable map breaks V into a direct sum of eigenspaces. Splitting a space is rarely the end goal; it is the *tool* for understanding a transformation one block at a time.