Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Isomorphism with the coordinate space

Builds on Coordinate vectorLinear isomorphism

Statement

Let FF be a field, let VV be a vector space over FF, and let B=(b1,…,bn)B = (b_1, \dots, b_n) be a basis of VV. Then the map V→Fn,v↦[v]B,V \to F^n, \qquad v \mapsto [v]_B, that sends each vector to its coordinate vector in BB, is a linear isomorphism. Its inverse sends (a1,…,an)(a_1, \dots, a_n) to a1b1+⋯+anbna_1 b_1 + \cdots + a_n b_n.

Remarks

The isomorphism depends on the choice of the basis. It sends bib_i to the standard basis vector eie_i. Through this map, each statement about FnF^n gives a statement about VV.

Reading path

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

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear isomorphism, definition, no evidence levelLinear isomorphismdefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionIsomorphism with the coordinate space, statement, no evidence levelIsomorphism with the …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 map definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Linear isomorphism definition
  7. Linear span definition
  8. Spanning set definition
  9. Basis definition
  10. Coordinate vector definition
  11. Isomorphism with the coordinate space statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Coordinate_vector, The standard representation
About this page
Tag059THXBRBB
URIm2://059THXBRBB
Kindstatement, proposition
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 059THXBRBB.
UsesCoordinate vector, Linear isomorphism

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