Dimension and Rank

The rank of a matrix is simply the number of independent directions its outputs can point in: 1 if every output lands on a line, 2 if they fill a plane, and so on. Think of it as how much the matrix really does, as opposed to how big it looks. The dimension of the null space gets its own name too, "nullity," because it measures what gets sent to zero (null). It's good for telling at a glance how much information a matrix keeps. The rank–nullity formula $\operatorname{rank}A + \dim\operatorname{Nul}A = n$ looks abstract, but it's just a count: $\operatorname{rank}A$ (directions that survive), $\dim\operatorname{Nul}A$ (directions crushed to zero) and $n$ (the directions you started with). Nothing gets lost in the accounting. It's simpler than it looks.

Why it adds up. The dimension of a subspace is the number of vectors in any basis, and you already know how to find bases from row reduction. Every column of $A$ either holds a pivot, which gives one basis vector for the column space, or corresponds to a free variable, which gives one basis vector for the null space. No column can do both, and none can do neither. So rank + nullity = number of columns.

Bounds. The rank of an $m\times n$ matrix is at most $\min(m, n)$. That makes sense: each pivot needs its own row and its own column.

Interpretation. A transformation from $\mathbb R^n$ "spends" its $n$ input dimensions: some survive as output directions (rank), and the rest collapse (nullity).

A common mistake. It's tempting to think rank plus nullity equals the number of rows, since the rows are what you row reduce. It's the number of columns: the null space lives in the input space $\mathbb R^n$, and $n$ is the number of columns.

Image compression keeps only the top few "rank-one" pieces of a picture's matrix; the rank tells you how much independent information a dataset really holds.