Define the rank and nullity of a linear map between finite-dimensional vector spaces.

State the rank-nullity formula.

Let $\alpha: U \rightarrow V$ and $\beta: V \rightarrow W$ be linear maps. Prove that

$$\operatorname{rank}(\alpha)+\operatorname{rank}(\beta)-\operatorname{dim} V \leqslant \operatorname{rank}(\beta \alpha) \leqslant \min \{\operatorname{rank}(\alpha), \operatorname{rank}(\beta)\}$$

Part IB