{wiki}

For a linear map $T:V\to W$ with finite-dimensional domain,
$$
\dim V=\dim\ker T+\dim\operatorname{im}T.
$$
If a basis of $\ker T$ is extended to a basis of $V$, the images of the added basis vectors form a basis of $\operatorname{im}T$. Counting the two parts proves the formula.


Rank-nullity theorem

{wiki=Rank–nullity_theorem}

The Rank-Nullity Theorem is a fundamental result in linear algebra that relates the dimensions of different subspaces associated with a linear transformation. Specifically, it applies to linear transformations between finite-dimensional vector spaces.


 Rank–nullity theorem

ID: rank-nullity-theorem