The range of the controllability matrix

Question

Consider a matrix $M$, the range of $M$, denoted by $R(M)$:

$R(M) = \{b | b = Mx\}.$

Now, consider the controllability matrix $$C = \begin{bmatrix}B&AB & \dots& A^{n-1}B\end{bmatrix}=\\= \begin{bmatrix}I& A&\dots&A^{n-1}\end{bmatrix}B$$

How to understand the columns of $B$ belongs to $R(C)$? I hope a more intuitive way to see this problem.

Thanks!

Answer 1

The range of a matrix is the span of the column vectors of that matrix. The columns of $B$ are also columns of $C$ by definition, so they are in $R(C)$.