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Definitions from Basis and dimension

All definitions from Basis and dimension.

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Basis

A set BB in a vector space VV is a basis for VV if it is linearly independent and spans VV.

Spanning gives a representation for every vector in VV and independence makes it unique, as shown in linear independence. A basis is therefore a generating set with no redundant direction: removing any basis vector destroys spanning.

For an infinite set, span still uses finite linear combinations and independence means that every finite subset is independent. The calculations below use finite bases.

  1. [1]:Span
  2. [2]:Linear independence
  3. [3]:Eigenbasis
  4. [4]:Diagonalisation
  5. [5]:A system with zero initial derivative
  6. [6]:Values on a basis determine a linear map
  7. 23 more

Dimension

If a vector space VV has a finite basis, then VV is finite-dimensional and its dimension, denoted dim⁡(V)\dim(V), is the common number of vectors in all bases of VV.

The zero vector space has the empty basis, so dim⁡({0})=0\dim(\{0\})=0. If VV has no finite basis, it is called infinite-dimensional. For example, P(R)P(\mathbb R) is infinite-dimensional: no finite list of polynomials can span polynomials of arbitrarily high degree.

  1. [1]:Basis
  2. [2]:All bases have the same size
  3. [3]:A system with zero initial derivative
  4. [4]:Nullity
  5. [5]:Rank
  6. [6]:Rank-nullity
  7. 8 more

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BasisDimension