Editing Controllability (section)

===Example===
For example, consider the case when <math>n=2</math> and <math>r=1</math> (i.e. only one control input). Thus, <math>B</math> and <math>A B</math> are <math>2 \times 1</math> vectors. If <math>\begin{bmatrix}B & AB\end{bmatrix}</math> has rank 2 (full rank), and so <math>B</math> and <math>AB</math> are [[linearly independent]] and span the entire plane. If the rank is 1, then <math>B</math> and <math>AB</math> are [[Line (geometry)|collinear]] and do not span the plane.

Assume that the initial state is zero.

At time <math>k=0</math>: <math>x(1) = A\textbf{x}(0) + B\textbf{u}(0) = B\textbf{u}(0)</math>

At time <math>k=1</math>: <math>x(2) = A\textbf{x}(1) + B\textbf{u}(1) = AB\textbf{u}(0) + B\textbf{u}(1)</math>

At time <math>k=0</math> all of the reachable states are on the line formed by the vector <math>B</math>.
At time <math>k=1</math> all of the reachable states are linear combinations of <math>AB</math> and <math>B</math>.
If the system is controllable then these two vectors can span the entire plane and can be done so for time <math>k=2</math>.
The assumption made that the initial state is zero is merely for convenience.
Clearly if all states can be reached from the origin then any state can be reached from another state (merely a shift in coordinates).

This example holds for all positive <math>n</math>, but the case of <math>n=2</math> is easier to visualize.