Intersection of subspaces is a subspace ↩

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created: 2021-11-06 00:57:52
modified: 2022-01-10 04:13:04

Statement:
If is a vector space and is any collection of subspaces, then:

is a subspace.

Proof

By definition, is a subspace if any sum of its vectors stays in and these vectors stay in when multiplied by any scalar.

  1. Now let . Then if we consider their sum, they must stay in each since they are subspaces as well: is part of every subspace, but then it must be part of as well the sum part of the definition holds.
  2. The same consideration is true for multiplying with scalars too.