Linear algebra MOC

Complement subspace

Let be a vector space over and be a subspace. A complement is a subspace such that the internal direct sum .

Properties

1. Every has a (in general not unique) complement .¹

Proof of 1.

The existence of the compliment follows from Every vector space has a basis: Let be a basis of . Then there exists a basis of such that . Then is a complement of .

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Footnotes

1. This is downstream from AC. ↩