The Rank–Nullity Theorem

In the setting of Linear Algebra: Standing Notation, let $K$ be a field, let $V$ be a finite-dimensional vector space over $K$, let $W$ be a vector space over $K$, and let $T$ be a linear map from $V$ to $W$.

(i) Rank–nullity. The image $\operatorname{im}T$ is finite-dimensional, the kernel $\ker T$ is finite-dimensional (Properties of Dimension (v)), and

$$\dim V=\dim\ker T+\dim\operatorname{im}T.$$