The dimension of a vector space V is the size of a basis for that vector space written:
dim Span = rank
If U is a subspace of W then
Example:
For any linear function <math>f : V \mapsto W</math> : <MATH> \text{dim } \href{function#kernel}{Ker} f + \text{dim } \href{function#image}{lm} f = \text{dim V} </MATH>
Apply Kernel-Image Theorem to a matrix function f (x) = Ax:
For any n-column matrix A, <MATH> \href{matrix#nullity}{nullity } A + \href{rank}{rank } A = n </MATH>
Example
Dual Dimension Theorem: <MATH>dim V + dim V^* = n</MATH>
Proof:
Let <math>a_1, \dots , a_m</math> be generators for V.
Let <math>A = \begin{bmatrix} \begin{array}{r} a_1 \\ \hline \\ \vdots \\ \hline \\ a_m \end{array} \end{bmatrix} </math> then <math>V^* = \href{matrix#null space}{Null} A</math>
Rank-Nullity Theorem states that