Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Dimension of the coordinate space

Builds on DimensionCoordinate space

Statement

Let FF be a field and let n≥0n \ge 0 be an integer. The coordinate space FnF^n has dimension nn over FF.

Remarks

The standard basis e1,…,ene_1, \dots, e_n is a basis of FnF^n with nn vectors. A consequence: each nn-dimensional space over FF is isomorphic to FnF^n. In Mathlib, the index type is Fin n, and the space is the type of functions Fin n → F.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Coordinate space, object, no evidence levelCoordinate spaceobjectVector 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 levelDimensiondefinitionDimension of the coordinate space, statement, no evidence levelDimension of the coor…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. Coordinate space object
  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. Basis definition
  9. Finite-dimensional vector space definition
  10. Dimension definition
  11. Dimension of the coordinate space statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dimension_(vector_space), Examples
About this page
Tag0QK7FPE6VE
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Kindstatement, proposition
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 0QK7FPE6VE.
UsesDimension, Coordinate space

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