Sums of Subspaces

The union of subspaces is rarely a subspace, which is why we shall define and work with the sum rather than unions.

Theorem 1.8.1. The union of two subspaces of a vector space V is a subspace of V if and only if one of the subspaces is contained in the other.

Proof. Suppose W₁ and W₂ are two subspaces of a vector space V over a field 𝔽.

If W₁ ⊆ W₂ then, W₁ ∪ W₂ = W₂ and hence W₁ ∪ W₂ is a subspace.
If W₂ ⊆ W₁ then, W₁ ∪ W₂ = W₁ and hence W₁ ∪ W₂ is a subspace.

Conversly, suppose W₁ ∪ W₂ is a subspace. We shall show that either W₁ ⊆ W₂ or W₂ ⊆ W₁.
Let us assume that neither W₁ is a subset of W₂ nor W₂ is a subset of W₁. Then

W₁ not a subset of W₂ ⇒ ∃α ∈ W₁ ∉ W₂ (1)
W₂ not a subset of W₁ ⇒ ∃β ∈ W₂ ∉ W₁ (2)

From (1) and (2) we have α ∈ W₁ ∪ W₂ and β ∈ W₁ ∪ W₂. Since by hyphotheis W₁ ∪ W₂ is a subspace, therefore α + β ∈ W₁ ∪ W₂. But α + β ∈ W₁ ∪ W₂ ⇒ α + β ∈ W₁ or α + β ∈ W₂. Suppose α + β ∈ W₁. Since α ∈ W₁ and W₁ is a subspace, therefore (α + β) − α = β is in W₁. But from (2) we have β ∉ W₁, thus we get a contradiction. Supposing that α + β ∈ W₂. Since β ∈ W₂ and W₂ is a subspace, therefore (α + β) − β = α is in W₂. But from (1) we have α ∉ W₂, thus we get again a contradiction. Hence either W₁ ⊆ W₂ or W₂ ⊆ W₁. □

Theorem 1.8.3. (Intersection Theorem) If U and W are subspaces of a vector space V, then so is U ∩ W.

Proof. First we observe that since U and W are subspaces, 0 ∈ U ∩ W. Suppose that in addition, x,y ∈ U ∩ W. Then x,y ∈ U and x,y ∈ W. Since U and W are subspaces x + y ∈ U and αx ∈ U for any scalar α. Similarly x + y ∈ W and αx ∈ W. Hence x + y ∈ U ∩ W and αx ∈ U ∩ W for all scalars α. □

Definition 1.8.4 (Sum of subsets) Suppose U₁, ..., U_m are subsets of V. The sum of U₁, ..., U_m, denoted U₁ + ... + U_m, is the set of all possible sums of elements of U₁, ..., U_m
More precisely,

U₁ + ... + U_m = {u₁ + ... + u_m: u₁ ∈ U₁, ..., u_m ∈ U_m} □

This definition extends to subspaces, for example U and V are subspaces of the vector space W, their sum is

U + W ={ u + v | u ∈ U, w ∈ W}

Example 1.8.2 Suppose U is the set of all elements of 𝔽³ whose second and third coordinates equal 0, and W is the set of all elements of 𝔽³ whose first and third coordinates equal 0:

U ={(x, 0, 0) ∈ 𝔽³ : x ∈ 𝔽} and W ={(0, y, 0) ∈ 𝔽³ : y ∈ 𝔽}

Then

U + W ={(x, y, 0): x,y ∈ 𝔽}

as you should verify. ■

Direct Sum

If two vector subspaces intersect trivially, that is U ∩ V = {0}, then the sum U + W is called a direct sum and is written as U ⊕ V.

Definition 1.8.3 (Internal Direct Sum). If U and V are subspaces of the vector space W such that

span(U ∪ V) = V, and
U ∩ V = {0}.

then the subspace U + V is called internal direct sum of subspaces, denoted by U ⊕ V. In this case U and V are called summands of U ⊕ V. □

We used the term, "internal" to distinguish it from the external direct product, introduced when groups were covered: this direct sum is the direct product U x V of two spaces with coordinate-wise arithmetic on elements of the product.
It is also worth noting that the defining property (a) of the direct sum is equivalent to saying that every vector v ∈ V can be written in the form w = u₁ + v₁ for some u₁ ∈ U and v₁ ∈ V.

Properties of the Direct Sum

A direct sum has considerably more interesting propertiese than simply the sum of two vector spaces; One is that dimensions simply add up

dim(U ⊕ V) = dim(U) + dim(V)

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