Math 2.0

Linear and multilinear algebra; matrix theory Proposition

A spanning set of R^m has at least m vectors

Builds on Spanning setEuclidean space

Statement

Let mm be a positive integer. Suppose a set of vectors of ℝm\mathbb{R}^m spans the whole of the space ℝm\mathbb{R}^m. Then the set has at least mm vectors.

Remarks

Equivalently, a set with fewer than mm vectors in ℝm\mathbb{R}^m never spans ℝm\mathbb{R}^m.

In matrix terms, the matrix whose columns are the vectors has mm rows. To span, it needs a pivot position in every row. Each column holds at most one pivot, so there must be at least mm columns.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectSpanning set, concept, no evidence levelSpanning setconceptA spanning set of R^m has at least m vectors, statement, no evidence levelA spanning set of R^m…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. Euclidean space object
  2. Spanning set concept
  3. A spanning set of R^m has at least m vectors statement

Further reading

  1. Web https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section2-3.xml#L1387, src/section2-3.xml:1387
About this page
TagD23BB74RYS
URIm2://D23BB74RYS
Kindstatement, proposition
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status D23BB74RYS.
AboutSpanning set
UsesSpanning set, Euclidean space
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.