Prove that if a linear system has infinitely many solutions, then any solution could be written as a linear function of free parameters, thanks.

Question

Suppose $Ax=b$, where $A$ is of dimension $q\times p$ with $q<p$, $rank(A)=q$, $b$ is $q \times 1$. Let $\mathcal{X}=\{x:Ax=b\}$ be the solution set of this system. How to rigorously prove that any solution $x^{*}\in \mathcal{X}$ could be written as $x^{*}=B\pi_{free}+w$ for some free parameter vector $\pi_{free}$ of dimension $(p-q)\times 1$, $B$ of dimension $p\times (p-q)$, $w$ of dimension $p\times 1$? Example: $x=\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}$, $A=\begin{bmatrix}1&0&-1\\0&1&-1\end{bmatrix}$, $b=\begin{bmatrix}1\\2\end{bmatrix}$, then any solution to this system could be written as $x^{*}=\begin{bmatrix}1\\1\\1\end{bmatrix}x_{free}+\begin{bmatrix}1\\2\\0\end{bmatrix}$ with $B=\begin{bmatrix}1\\1\\1\end{bmatrix}$, $x_{free}=x_{3}$, and $w=\begin{bmatrix}1\\2\\0\end{bmatrix}$. Thank you very much!