31. Dimension
The dimension of a vector space is the number of vectors in any basis of .
This number doesn’t depend on which basis you pick — every basis of the same vector space has the same size.
Example
- (the standard basis has vectors)
- of the -plane in is
- (the trivial space has the empty basis)
- of the space of matrices is
31.1. Properties
For subspaces :
- implies (no proper subspace of full dimension)
- (Grassmann formula)
-
For a linear transformation :
31.2. Computing dimensions of common subspaces
For an matrix :
- (the nullity)
- (column rank = row rank)
31.3. See also
- Basis — what dimension counts
- Rank — dimension of column / row / image
- Rank–Nullity Theorem