Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Linearly independent set of the right size is a basis

Builds on BasisDimensionLinear independence

Statement

Let FF be a field and let VV be a vector space over FF of finite dimension nn. Let v1,…,vn∈Vv_1, \dots, v_n \in V be nn vectors. If v1,…,vnv_1, \dots, v_n are linearly independent, then they form a basis of VV. If v1,…,vnv_1, \dots, v_n span VV, then they form a basis of VV.

Remarks

Thus, for exactly dim⁡V\dim V vectors, independence and spanning are equivalent. For example, three vectors of ℝ3\R^3 form a basis if and only if the 3×33 \times 3 matrix with these columns has nonzero determinant. More generally, a family v1,…,vkv_1, \dots, v_k is linearly independent if and only if its span has dimension kk.

Reading path

To understand this page from the ground up, read these 10 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 levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionLinearly independent set of the right size is a basis, statement, no evidence levelLinearly independent …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. Finite-dimensional vector space definition
  9. Dimension definition
  10. Linearly independent set of the right size is a basis statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Basis_(linear_algebra), Properties
About this page
Tag096V7H13H9
URIm2://096V7H13H9
Kindstatement, proposition
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 096V7H13H9.
UsesBasis, Dimension, Linear independence

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