Subsection DVS Dimension of Vector Spaces

We can now collect the dimension of some common, and not so common, vector spaces.

Theorem DCM Dimension of $\complex{m}$

The dimension of $\complex{m}$ (Example VSCV) is $m$.

Proof
Theorem DP Dimension of $P_n$

The dimension of $P_{n}$ (Example VSP) is $n+1$.

Proof
Theorem DM Dimension of $M_{mn}$

The dimension of $M_{mn}$ (Example VSM) is $mn$.

Proof
Example DSM22 Dimension of a subspace of $M_{22}$
Example DSP4 Dimension of a subspace of $P_4$
Example DC Dimension of the crazy vector space

It is possible for a vector space to have no finite bases, in which case we say it has infinite dimension. Many of the best examples of this are vector spaces of functions, which lead to constructions like Hilbert spaces. We will focus exclusively on finite-dimensional vector spaces. OK, one infinite-dimensional example, and then we will focus exclusively on finite-dimensional vector spaces.

Example VSPUD Vector space of polynomials with unbounded degree
Sage D Dimension
(in context)