Linear algebra MOC

Vector subspace

A vector subspace of a vector space is a subset that is a vector space under the same scalar multiplication and vector addition. #m/def/linalg This can be boiled down to the following requirement:

If and , then .

The concept of subspaces naturally leads to the concept of a Span, which is the smallest possible subspace containing a set of specific vectors within the main vector space.

Properties

  1. The subspaces of a given vector space form a Complete lattice with initial and terminal . The greatest lower bound is the intersection of subspaces, the least upper bound is the sum of subspaces.
  2. A nontrivial vector space over an infinite field is not the union of finitely many proper subspaces.1
Proof of 2

Let be a nontrivial vector space over . Assume and without loss of generality . Now let and . Then the infinite set

is an infinite set corresponding to the line through parallel to . We will show that contains at most one element from each and must thence be finite, leading to contradiction.

First note that if for then since , contradicting our assumption. Next, suppose for some we have and , Then

so , which is also a contradiction.

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

Footnotes

  1. 2008. Advanced Linear Algebra, p. 39 ↩