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Direct Sum — Definition, Formula & Examples

Direct sum is a way of combining two or more vector spaces (or subspaces) into a larger one where each element can be written uniquely as a sum of elements from the component spaces.

Given vector spaces V1V_1 and V2V_2 over a field FF, the direct sum V1⊕V2V_1 \oplus V_2 is the vector space of all ordered pairs (v1,v2)(v_1, v_2) with v1∈V1v_1 \in V_1 and v2∈V2v_2 \in V_2, equipped with componentwise addition and scalar multiplication. For subspaces W1,W2W_1, W_2 of a vector space VV, we write V=W1⊕W2V = W_1 \oplus W_2 when V=W1+W2V = W_1 + W_2 and W1∩W2={0}W_1 \cap W_2 = \{\mathbf{0}\}.

Key Formula

V=W1⊕W2  ⟺  V=W1+W2 and W1∩W2={0}V = W_1 \oplus W_2 \iff V = W_1 + W_2 \text{ and } W_1 \cap W_2 = \{\mathbf{0}\}
Where:
  • VV = The ambient vector space
  • W1,W2W_1, W_2 = Subspaces of V
  • ⊕\oplus = Direct sum symbol

How It Works

When you form the direct sum V1⊕V2V_1 \oplus V_2, the dimension of the result equals dim⁡(V1)+dim⁡(V2)\dim(V_1) + \dim(V_2). Every vector in the direct sum decomposes uniquely into one component from each summand — this uniqueness is what distinguishes a direct sum from an ordinary sum of subspaces. You can recognize an internal direct sum by checking two conditions: the subspaces together span the whole space, and their intersection is only the zero vector.

Worked Example

Problem: Let W₁ = {(a, 0) : a ∈ ℝ} and W₂ = {(0, b) : b ∈ ℝ} be subspaces of ℝ². Show that ℝ² = W₁ ⊕ W₂.
Step 1: Check that W₁ + W₂ = ℝ². Any vector (x, y) in ℝ² can be written as (x, 0) + (0, y), where (x, 0) ∈ W₁ and (0, y) ∈ W₂.
(x,y)=(x,0)+(0,y)(x, y) = (x, 0) + (0, y)
Step 2: Check that W₁ ∩ W₂ = {(0, 0)}. A vector in both W₁ and W₂ must have the form (a, 0) and (0, b) simultaneously, forcing a = 0 and b = 0.
W1∩W2={(0,0)}W_1 \cap W_2 = \{(0,0)\}
Step 3: Both conditions are satisfied, so ℝ² is the direct sum of W₁ and W₂. Note that dim(W₁) + dim(W₂) = 1 + 1 = 2 = dim(ℝ²).
R2=W1⊕W2\mathbb{R}^2 = W_1 \oplus W_2
Answer: ℝ² = W₁ ⊕ W₂, confirming that every vector in ℝ² decomposes uniquely into a horizontal and vertical component.

Why It Matters

Direct sums appear throughout linear algebra whenever you decompose a space into independent pieces — for instance, splitting ℝⁿ into eigenspaces of a diagonalizable matrix. In abstract algebra, the same idea generalizes to modules and groups, making it foundational for courses in representation theory and algebraic topology.

Common Mistakes

Mistake: Confusing a direct sum with an ordinary sum of subspaces. Students write V = W₁ ⊕ W₂ whenever W₁ + W₂ = V, ignoring the intersection condition.
Correction: A direct sum requires both that the subspaces span V and that their intersection is {0}. Without the trivial intersection, decompositions are not unique, and the sum is not direct.

Related Terms