Rank-one updates intuition

A rank-one matrix is an outer product $\mathbf{u}\mathbf{v}^T$: one column direction times one row direction . Every rank-one map squashes all inputs onto a line (or collapses further). Adding such a matrix perturbs column and null structures in controlled ways.

Rank satisfies $\mathrm{rank}(A+B)\le \mathrm{rank}(A)+\mathrm{rank}(B)$: combining maps cannot create more independent directions than the sum of the parts. Sherman-Morrison formulas in optimization and statistics exploit rank-one updates to invert efficiently after a small change.

If $A$ has rank $r$ on domain $\mathbb{R}^n$, nullity is $n-r$ by rank-nullity. That count is how many invisible inputs you can add to a solution without changing $\mathbf{b}$.

Example inconsistency: columns both along $(1,1)$ but $\mathbf{b}=(1,0)$ lies off that line in $\mathbb{R}^2$. Rank 1 means a thin reachable set. Sherman-Morrison style updates appear when a rank-one correction is added to an already inverted matrix; the formulas are worth meeting after you trust column and null space language. Rank bounds how much a sum of matrices can increase dimension: $\mathrm{rank}(A+B)\le \mathrm{rank}A+\mathrm{rank}B$.