Let $U, V$ be finite-dimensional vector spaces, and let $\theta$ be a linear map of $U$ into $V$. Define the rank $r(\theta)$ and the nullity $n(\theta)$ of $\theta$, and prove that

$$r(\theta)+n(\theta)=\operatorname{dim} U$$

Now let $\theta, \phi$ be endomorphisms of a vector space $U$. Define the endomorphisms $\theta+\phi$ and $\theta \phi$, and prove that

$$\begin{aligned} r(\theta+\phi) & \leqslant r(\theta)+r(\phi) \\ n(\theta \phi) & \leqslant n(\theta)+n(\phi) . \end{aligned}$$

Prove that equality holds in both inequalities if and only if $\theta+\phi$ is an isomorphism and $\theta \phi$ is zero.