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Section3.7Sums and Intersections of Subspaces

Objectives
  1. Understand sums and intersections of subspaces.
  2. Theorem: Grassmann's formula.
  3. Vocabulary: sum, intersection of subspaces.

In this section we discuss sums and intersections of subspaces, U+V and U∩V, and how to find bases for them given bases for U and V.

Definition

Suppose we're given subspaces U and V of Rn. Then their sum is the subspace consisting of sums u+v where u∈U and v∈V:

U+V={u+v|u∈U,v∈V}⊆Rn

It is easy to see that U+V is indeed a subspace of Rn (exercise). If U=Col(A) and V=Col(B), i.e., if U and V are given as spans of the columns of matrices A and B, respectively, then U+V=ColAABB where the matrix AABB is obtained by putting A and B side by side. In other words, to get a spanning set for U+V we take the union of spanning sets for U and V

Definition

Suppose we're given subspaces U and V of Rn. Then their intersection is the subspace consisting of vectors that belong to both U and V:

U∩V={x∈Rn|x∈U∧x∈V}⊆Rn

Again, it's easy to see that U∩V is a subspace of Rn. This time, we have the dual behavior as for sums: If U=Nul(A) and V=Nul(B), i.e., if U and V are given as null spaces of matrices A and B, respectively, then U∩V=NulCABD, where the matrix CABD is obtained by putting A and B on top of each other. In other words, if U and V are solution sets to homogeneous systems of linear equations, then U∩V is the solution set to the combined system of all the homogeneous linear equations involved.

Proof
Example