Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Intersection of subspaces

Builds on Linear subspace

Statement

Let FF be a field, let VV be a vector space over FF, and let UU and WW be linear subspaces of VV. Then U∩WU \cap W is a linear subspace of VV. It is the largest subspace of VV that is contained in both UU and WW. More generally, the intersection of each nonempty family of subspaces of VV is a subspace of VV.

Remarks

The union U∪WU \cup W is a subspace only when U⊆WU \subseteq W or W⊆UW \subseteq U. The smallest subspace that contains U∪WU \cup W is the sum U+WU + W. In ℝ3\R^3, two different planes through the origin meet in a line through the origin.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionIntersection of subspaces, statement, no evidence levelIntersection of subsp…statement
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  1. Vector space definition
  2. Linear subspace definition
  3. Intersection of subspaces statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Linear_subspace, Operations and relations on subspaces
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MSC15A03
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