Explain: Independent Column variables as Linear combination of free variables.

Question

I am reading a book on control systems and stuck on a text in it.

We have $Sx = 0$ where S $\in$ $\Re^{m x n}$ and is full rank i.e Rank of S = m. $x \in \Re^n$

Now it states that

"Exactly m of the states can be expressed as a linear combination of the remaining n-m states."

I am not getting hold of this. I know there are n-m free variables, but how can the other variables be expressed as linear combination of free variables?

Answer 1

It might help to consider a concrete example. Take $m = 2 < 5 = n$ and let: $$S = \begin{bmatrix} 1 & 2 & 0 & 3 & 4 \\ 0 & 0 & 1 & 5 & 6 \end{bmatrix}$$ which has full rank (it has the maximum number of pivots, $m = 2$).

Observe that exactly $m = 2$ of the basic variables (namely, $x_1, x_3$) can be expressed as a linear combination of the remaining $n - m = 3$ free variables (namely, $x_2, x_4, x_5$) since: \begin{cases} x_1 = -2x_2 - 3x_4 - 4x_5 \\ x_3 = -5x_4 - 6x_5 \end{cases}