Full conceptual checklist before nonsquare focus

You now separate existence (column space membership) from uniqueness (null space size) . Square invertible maps are the happy case where both succeed. Tall maps $m\gt n$ with full column rank are injective but usually not surjective onto $\mathbb{R}^m$. Wide maps with full row rank are surjective onto $\mathbb{R}^m$ but typically have many solutions when consistent.

Row space dimension equals rank; so does column space dimension. Transpose swaps row and column roles in the dual picture even before the dedicated dot-product chapter makes that geometric.

Next chapter relaxes square shape while keeping rank language. Express reachability without jargon: $\mathbf{b}$ is some output $A\mathbf{x}$ for at least one input $\mathbf{x}$. Tall full-column-rank maps are injective but usually not surjective when $m\gt n$; wide full-row-rank maps are surjective onto $\mathbb{R}^m$ but have nontrivial null spaces when $m\lt n$. Row space dimension also equals rank, linking row and column pictures . Chapter 8 relaxes square shape while keeping this existence versus uniqueness split. Express reachability as: $\mathbf{b}$ equals $A\mathbf{x}$ for some $\mathbf{x}$. Tall injective maps still miss most outputs when $m\gt n$.