I'm reading about geographical tomography. Consider the following text:
The tomography leads to a linear system $$Ax = b$$ with $A\in\mathbb{R}^{m\times n}$, $x\in\mathbb{R}^n$ and $b\in\mathbb{R}^m$. Solutions to this problem satisfy the normal equations $$A^TAx_{LS} = A^Tb$$ Note a couple of things:
- The system may be inconsistent ($b\notin\mathcal{R}(A)).$
- Suppose that $rank(A) < n$ and $x_{LS}$ is a least-squares solution. Then $$\hat{x} = x_{LS} + y\text{ with }y\in\mathcal{N}(A)$$ is also a least squares solution.
What do the notations $\mathcal{R}$ and $\mathcal{N}$ mean? I've never encountered these before.
It looks like they're using $\mathcal{R}(A)$ and $\mathcal{N}(A)$ as column space of $A$ and the null space of $A$, respectively.
The column space of a matrix is all linear combinations of the columns of the matrix, which is all possible values of $Ax$ for some vector $x$. So, $Ax = b$ has no solution if $b \notin \mathcal{R}(A)$.
The null space of a matrix is all solutions of $Ax = 0$. So if $y \in \mathcal{N}(A)$, then $$A^TA(x_{LS} + y) = A^TAx_{LS} + A^TAy = A^TAx_{LS} + 0 = A^TAx_{LS} = A^Tb$$