In Jousef Saad's book (see ref.1), the author introduces the projection method capable of solving a large set of linear system of equations iteratively.
Below is what extracted from the book about the method:
For linear system of equations $$\boldsymbol{A} \boldsymbol{x} = \boldsymbol{b}$$ where $\boldsymbol{A} \in \mathbb{R}^{n \times n}$, $\boldsymbol{x}$ is the vector of solution ($\boldsymbol{x} \in \mathbb{R}^{n}$).
A projection method onto $\mathcal{K}_m$ and orthogonal to $\mathcal{L}_m$ is a process which finds an approximate solution $\tilde{\boldsymbol{x}}\in \mathbb{R}^{n}$ satisfying
$$ \mathrm{Find\ } \tilde{\boldsymbol{x}}\in \mathcal{K}_m,\quad \mathrm{such\ that}\quad \boldsymbol{b} -\boldsymbol{A}\tilde{\boldsymbol{x}} \bot \mathcal{L}_m $$ where $\mathcal{K}_m$ and $\mathcal{L}_m$ are two $m$-dimensional subspace of $\mathbb{R}^n$ ($m \leqslant n$), which are termed the search subspace and the constraint subspace, respectively.
Given an initial guess $\boldsymbol{x_0}$ to the solution, the approximation must be sought in an affine space $\boldsymbol{x_0}+\mathcal{K}_m$ instead of the homogeneous vector space $\mathcal{K}_m$.
$$ \mathrm{Find\ } \tilde{\boldsymbol{x}}\in \boldsymbol{\tilde{x}_0} + \mathcal{K}_m,\quad \mathrm{such\ that}\quad \boldsymbol{b} - \boldsymbol{A}\tilde{\boldsymbol{x}} \bot \mathcal{L}_m $$
To clarify my question below, I take a low-dimensional case in the 3-D cubic space ($n = 3$, $m = 2$) as an example and sketch a illustration:
Illustration of subspaces of a 3-D space in the projection method
Questions:
What is the geometric significance of the affine space $\boldsymbol{x_0}+\mathcal{K}_m$, or how does it look like in the cubic?
What is the spatial relationship between $\mathcal{K}_m$ and the affine space $\boldsymbol{x_0}+\mathcal{K}_m$ in the cubic?
Reference:
Saad, Yousef, 2003, Iterative methods for sparse linear systems., Philadelphia, PA: SIAM Society for Industrial and Applied Mathematics. xviii, p. 134 .
