Dimension

Given a subspace, there can be infinitely many bases. If this is not intuitive, see span and basis.

Definition

If is a basis for a subspace of vector space , then we say the dimension of is k, and we write .

That is, . (see cardinality)

Intuition

It takes 2 2-dimensional vectors to span , and 3 3-dimensional vectors to span .

Dimension of Fundamental Subspaces

Theorem

Let

(see rank)

Intuition

The rank of a matrix is the number of dimensions the column space can span.

If it's easier, the use of "column space" can be taken to mean each vector in the matrix.