Math 2.0

Linear and multilinear algebra; matrix theory Proposition

A linearly independent set in R^m has at most m vectors

Builds on Linear independenceSteinitz exchange lemma

Statement

Let mm be a positive integer. Suppose a set of vectors in ℝm\mathbb{R}^m is linearly independent. Then the set has at most mm elements.

Remarks

Equivalently, any set of more than mm vectors in ℝm\mathbb{R}^m is linearly dependent. This is a special case of the Steinitz exchange lemma, since the mm standard unit vectors span ℝm\mathbb{R}^m.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first. The path takes the proof Proof of the Steinitz exchange lemma by induction.

Show the path as a graph
Linear independence, concept, no evidence levelLinear independenceconceptVector 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 setdefinitionProof of the Steinitz exchange lemma by induction, proof, no evidence levelProof of the Steinitz…proofSteinitz exchange lemma, statement, no evidence levelSteinitz exchange lem…statementA linearly independent set in R^m has at most m vectors, statement, no evidence levelA linearly independen…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. Linear independence concept
  2. Vector space definition
  3. Linear combination definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Linear span definition
  7. Spanning set definition
  8. Proof of the Steinitz exchange lemma by induction proof
  9. Steinitz exchange lemma statement
  10. A linearly independent set in R^m has at most m vectors statement

Further reading

  1. Web https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section2-4.xml#L777, src/section2-4.xml:777
About this page
TagXCG7M9ZHP2
URIm2://XCG7M9ZHP2
Kindstatement, proposition
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status XCG7M9ZHP2.
UsesLinear independence, Steinitz exchange lemma
Changed4 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.