I was looking around for a table detailing the difference in solving/approximating linear systems in under- and over-determined cases but didn't find anything that summarized the cases nicely. I'm trying to fill out the following table for use as a teaching/studying device and have pulled in some details from Wikipedia etc. Any additions/suggestions would be appreciated!
$$ \begin{aligned} &\text { Linear Systems Summary: } A \in \mathbb{R}^{m \times n}, x \in \mathbb{R}^{n}, \text { and } b \in \mathbb{R}^{m}\\ &\begin{array}{lllllll} \hline \text { type } & \text { size } & \text { case } & \text { rank(A) } & \text { true solns } & A^{\dagger} \text { soln } & \text { comments } \\ \hline \text { Overdetermined } & m>n & b \notin \mathcal{R}(A) & \text { full col } & \begin{array}{l} \text { none } \\ \text { (inconsistent) } \end{array} & \min A x-b_{2}^{2} & \begin{array}{l} \text { "least-squares" } \\ \text { problem } \end{array} \\ \hline \text { Overdetermined } & m>n & b \notin \mathcal{R}(A) & <n & \text { none } & & \\ \hline \text { Overdetermined } & m>n & b \in \mathcal{R}(A) & & & & \\ \hline \text { Underdetermined } & m<n & b \in \mathcal{R}(A) & \text { full row } & \infty & \text{min } x & \text { "least-norm" } \\ & & & & x^{*}=A^{\dagger} b+\left[I-A^{\dagger} A\right] w & \text { s.t. } A x=b & \text { problem } \\ \hline \text { Underdetermined } & m<n & b \in \mathcal{R}(A) & <m & \infty & & \\ \hline \text { Underdetermined } & m<n & b \notin \mathcal{R}(A) & \text { full row } & \infty & & \\ \hline \end{array} \end{aligned} $$