Math 2.0

statement theorem

Finite-dimensional spaces are isomorphic iff they have equal dimension

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.

Explanations

L1 human

Up to isomorphism, a finite-dimensional vector space is determined by one number: its dimension. All three-dimensional real spaces behave like R3\R^3. Only the names of the vectors differ.

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To read this node, read these 12 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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  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

Sources

  1. Web https://en.wikipedia.org/wiki/Dimension_(vector_space), Properties

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