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Rank and nullity
The dimensions of the image and kernel summarize how a linear transformation redistributes the dimensions of its input space.
The rank of $T$ is the dimension of its image:
$$\operatorname{rank}T=\dim(\operatorname{im}T).$$
The nullity is the dimension of its kernel:
$$\operatorname{nullity}T=\dim(\ker T).$$
Rank measures reachable directions
If a matrix transformation has rank $r$, its outputs span an $r$-dimensional subspace. For a matrix, the rank is also the number of pivot columns and the dimension of its column space.
Nullity measures lost freedom
The nullity counts independent directions in the domain that are sent to zero. In a homogeneous system
$$A\mathbf x=\mathbf0,$$
it is the number of independent parameters needed to describe all solutions.
Rank-nullity theorem
For a linear transformation with finite-dimensional domain $V$,
$$\dim V=\operatorname{rank}T+\operatorname{nullity}T.$$
For example, if $T:\mathbb R^3\to\mathbb R^2$ has a one-dimensional kernel, then
$$3=\operatorname{rank}T+1,$$
so its rank is $2$.
The theorem expresses a conservation of dimension: every independent input direction is either retained as an independent output direction or lost into the kernel.