Table of Contents
#source Axler "Linear Algebra Done Right" chapter 2.B
#flo #ref #disorganized #incomplete
1 Bases
1.1 Summary
If it spans, and it's linearly independent, it's a basis!
1.2 Axler2.27 #definition basis
A basis of \(V\) is a list of vectors in \(V\) that is linearly independent and spans \(V\). - Basically a linearly independent spanning list, or the "minimum" amount of information contained in a vector space
1.2.1 Other Results
- Axler2.29 "criterion for a basis"
- A list is a basis if and only if each vector in \(V\) can be written as exactly one linear combination of the list
- Axler2.31 all spanning lists contain a basis
- Intuitive. A spanning list might not be linearly independent, but some subset of it must be.
- Axler2.32 Any finite dimensional vector space has a basis
- Intuitive. It has a spanning list
- Also, no infinite dimensional vector space has a basis, by definition
- Axler2.33 Linearly indepedent lists can be extended to a basis
- Intuitive. Do this by adding in vectors that "bring new information"
- Axler2.34 Every subspace of \(V\) is part of a direct sum of \(V\)
- Intuitive. Kind of like saying there's an additive complement to every subspace of \(V\)
- Any vector space can be thought of the span of it's basis. Because \(V\) has a basis, and one of \(U\)'s basises can be written as a subsequence of \(V\)'s basis, that basis can be expanded and the expanded elements spanned to form the complement vecspace.