State and prove the Rank-Nullity theorem.

Let $\alpha$ be a linear map from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ of rank 2 . Give an example to show that $\mathbb{R}^{3}$ may be the direct sum of the kernel of $\alpha$ and the image of $\alpha$, and also an example where this is not the case.