Surjectivity: Hitting Every Output

A linear map $A: \mathbb{R}^n \to \mathbb{R}^m$ is surjective (or onto) if every vector in $\mathbb{R}^m$ is the output of some input: for every $\mathbf{b} \in \mathbb{R}^m$, the equation $A\mathbf{x} = \mathbf{b}$ has at least one solution.

Geometrically: the map "covers" all of $\mathbb{R}^m$ — no output is unreachable. This requires the columns of $A$ to span all of $\mathbb{R}^m$, which means $\text{Rank}(A) = m$.

Surjectivity requires at least as many columns as rows: $n \geq m$. A "fat" matrix ($n > m$) can be surjective; a square matrix may or may not be; a "tall" matrix ($n < m$) can never be surjective (too few columns to span $\mathbb{R}^m$).