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Mathematics LibreTexts

9.5: Sums and Intersections

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Outcomes

  1. Show that the sum of two subspaces is a subspace.
  2. Show that the intersection of two subspaces is a subspace.

We begin this section with a definition.

Definition 9.5.1: Sum and Intersection

Let V be a vector space, and let U and W be subspaces of V. Then

  1. U+W={u+w | uU and wW} and is called the sum of U and W.
  2. UW={v | vU and vW} and is called the intersection of U and W.

Therefore the intersection of two subspaces is all the vectors shared by both. If there are no vectors shared by both subspaces, meaning that UW={0}, the sum U+W takes on a special name.

Definition 9.5.2: Direct Sum

Let V be a vector space and suppose U and W are subspaces of V such that UW={0}. Then the sum of U and W is called the direct sum and is denoted UW.

An interesting result is that both the sum U+W and the intersection UW are subspaces of V.

Example 9.5.1: Intersection is a Subspace

Let V be a vector space and suppose U and W are subspaces. Then the intersection UW is a subspace of V.

Solution

By the subspace test, we must show three things:

  1. 0UW
  2. For vectors v1,v2UW,v1+v2UW
  3. For scalar a and vector vUW,avUW

We proceed to show each of these three conditions hold.

  1. Since U and W are subspaces of V, they each contain 0. By definition of the intersection, 0UW.
  2. Let v1,v2UW,. Then in particular, v1,v2U. Since U is a subspace, it follows that v1+v2U. The same argument holds for W. Therefore v1+v2 is in both U and W and by definition is also in UW.
  3. Let a be a scalar and vUW. Then in particular, vU. Since U is a subspace, it follows that avU. The same argument holds for W so av is in both U and W. By definition, it is in UW.

Therefore UW is a subspace of V.

It can also be shown that U+W is a subspace of V.

We conclude this section with an important theorem on dimension.

Theorem 9.5.1: Dimension of Sum

Let V be a vector space with subspaces U and W. Suppose U and W each have finite dimension. Then U+W also has finite dimension which is given bydim(U+W)=dim(U)+dim(W)dim(UW)

Notice that when UW={0}, the sum becomes the direct sum and the above equation becomes dim(UW)=dim(U)+dim(W)


This page titled 9.5: Sums and Intersections is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Ken Kuttler (Lyryx) via source content that was edited to the style and standards of the LibreTexts platform.

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