Math 2.0

Linear and multilinear algebra; matrix theory Vector spaces Theorem

Every linearly independent set extends to a basis

Builds on BasisLinear independence

Statement

Let FF be a field, let VV be a vector space over FF, and let L⊆VL \subseteq V be a linearly independent set. Then there is a basis BB of VV with L⊆BL \subseteq B.

Remarks

For finite-dimensional VV, add to LL, one at a time, a vector that is not in the span of the current set. This stops after finitely many steps. In general, the proof uses Zorn's lemma. A stronger form: if L⊆SL \subseteq S, with LL independent and SS spanning, then a basis BB exists with L⊆B⊆SL \subseteq B \subseteq S. For example, {(1,1,0)}\{(1, 1, 0)\} extends to the basis {(1,1,0),(0,1,0),(0,0,1)}\{(1, 1, 0), (0, 1, 0), (0, 0, 1)\} of ℝ3\R^3.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionEvery linearly independent set extends to a basis, statement, no evidence levelEvery linearly indepe…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Vector space definition
  2. Linear combination definition
  3. Linear subspace definition
  4. Linear independence definition
  5. Linear span definition
  6. Spanning set definition
  7. Basis definition
  8. Every linearly independent set extends to a basis statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Basis_(linear_algebra), Properties
About this page
Tag0M4SXE5WW3
URIm2://0M4SXE5WW3
Kindstatement, theorem
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 0M4SXE5WW3.
UsesBasis, Linear independence

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