Matrix equation from Ax=b to x=P+aQ

Question

I have an equation which describes a line, plane or higher dimensional object in the form.

$Ax=b$

So for example:

$\begin{bmatrix}1 & 1\end{bmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix} = 1$

This is a line through the points $(0,1)$ and $(1,0)$. How can i systematically write it to the form $P+\alpha Q=x$:

$\begin{bmatrix}1 \\ 0\end{bmatrix} + \alpha \begin{bmatrix}1 \\ -1\end{bmatrix} = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$

I am looking on how to do it with matrix operations only, so i can do it systematically, also with higher-dimensional equations.

Answer 1

If $Ax=b$ we are dealing with vector spaces, thus let's consider a basis for [x]:$[v_1], [v_2],...,[v_m]$

you can find $\alpha_i \in \mathbb{R}$, $i=1,m$ such that:

$\begin{bmatrix}x_1 \\|\\ x_m\end{bmatrix}=\alpha_1\begin{bmatrix}v_{11} \\|\\ v_{1m}\end{bmatrix} +...+ \alpha_m\begin{bmatrix}v_{n1} \\|\\ v_{nm}\end{bmatrix}$

thus $P=0$.

Whereas $P+\alpha Q=x$ is about affine spaces.

https://en.wikipedia.org/wiki/Affine_space