State the Rank-Nullity Theorem.

If $\alpha: V \rightarrow W$ and $\beta: W \rightarrow X$ are linear maps and $W$ is finite dimensional, show that

$$\operatorname{dim} \operatorname{Im}(\alpha)=\operatorname{dim} \operatorname{Im}(\beta \alpha)+\operatorname{dim}(\operatorname{Im}(\alpha) \cap \operatorname{Ker}(\beta))$$

If $\gamma: U \rightarrow V$ is another linear map, show that

$$\operatorname{dim} \operatorname{Im}(\beta \alpha)+\operatorname{dim} \operatorname{Im}(\alpha \gamma) \leqslant \operatorname{dim} \operatorname{Im}(\alpha)+\operatorname{dim} \operatorname{Im}(\beta \alpha \gamma)$$