Complement subspace

Let $𝑉$ be a vector space over $𝕂$ and $𝑈 \leq 𝑉$ be a subspace. A complement $𝑈 𝑐 \leq 𝑉$ is a subspace such that the internal direct sum $𝑈 \oplus 𝑈 𝑐 = 𝑉$ .

Properties

Every $𝑈 \leq 𝑉$ has a (in general not unique) complement $𝑈 𝑐 \leq 𝑉$ .¹

Proof of 1.

The existence of the compliment follows from Assuming choice, every vector space has a basis: Let $A$ be a basis of $𝑈$ . Then there exists a basis $A$ of $𝑉$ such that $A \subseteq B$ . Then $𝑈 𝑐 = s p a n (B ∖ A)$ is a complement of $𝑈$ .

#state/tidy | #lang/en | #SemBr

This is downstream from AC. ↩

Complement subspace

Properties

Footnotes