Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Finite-dimensional spaces are isomorphic iff they have equal dimension

Builds on Linear isomorphismDimension

Statement

Let FF be a field and let VV and WW be finite-dimensional vector spaces over FF. Then VV and WW are isomorphic if and only if dim⁡V=dim⁡W\dim V = \dim W, where dim⁡\dim is the dimension.

Remarks

An isomorphism sends a basis to a basis, so it keeps the dimension. Conversely, a bijection between bases of VV and WW extends to a linear isomorphism. The statement holds for all vector spaces, with dimension as a cardinal number. For example, M2×3(F)M_{2 \times 3}(F), F6F^6 and F[x]<6F[x]_{<6} are pairwise isomorphic.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
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 levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionFinite-dimensional spaces are isomorphic iff they have equal dimension, statement, no evidence levelFinite-dimensional sp…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. Finite-dimensional vector space definition
  11. Dimension definition
  12. Finite-dimensional spaces are isomorphic iff they have equal dimension statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dimension_(vector_space), Properties
About this page
Tag0VJBSK15SN
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Kindstatement, theorem
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 0VJBSK15SN.
UsesLinear isomorphism, Dimension

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