Math 2.0

definition

Dimension

Definition

Let FF be a field and let VV be a finite-dimensional vector space over FF. The dimension of VV over FF, written dim⁡FV\dim_F V or dim⁡V\dim V, is the number of vectors in a basis of VV. For an arbitrary vector space, the dimension is the cardinality of a basis.

Remarks

The number is well defined, by the dimension theorem for vector spaces. The zero space {0}\{0\} has the empty basis and dimension 00. The dimension depends on the field: dim⁡CC=1\dim_\C \C = 1, but dim⁡RC=2\dim_\R \C = 2. Mathlib writes Module.finrank F V for the dimension as a natural number, with value 00 for infinite-dimensional spaces.

Explanations

L1 human

The dimension counts the independent directions in a space. A line has dimension 11, a plane has dimension 22, and ordinary space has dimension 33. Every basis of a space has this same number of vectors.

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To read this node, read these 9 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 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 levelDimensiondefinition
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 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

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23 nodes use this node, directly or through other nodes.

Sources

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

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