Math 2.0

statement proposition

Dimension of the coordinate 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.

Explanations

L1 human

The space of lists of nn numbers has dimension nn. The nn standard unit vectors form a basis. For example, R3\R^3 has dimension 33.

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What this needs

To read this node, read these 11 nodes in this order. The foundations come first, and the list ends with this node.

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 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.
  • 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

Sources

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

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