How do I show that the two sequences below are short exact sequences of vector spaces?

Question

1. $0 \rightarrow \mathbb{Z_2}^k \to \mathbb{Z_2}^n \to \mathbb{Z_2}^{n-k} \to 0$

2. $0 \to \mathbb{Z_2} \to \mathbb{F}_{2^2} \to \mathbb{Z_2} \to 0$, where $\mathbb{F}_{2^2}$ is the Galois field of size $2^2$.

For the first, I have defined $\alpha : \mathbb{Z_2}^{k} \rightarrow \mathbb{Z_2}^n$ as $(a_1, \dots, a_k) \mapsto (0, \dots, 0, a_1, \dots a_k)$ and $\beta : \mathbb{Z_2}^n\to \mathbb{Z_2}^{n-k}$ as $(b_1, \dots, b_n) \mapsto (b_1, \dots, b_{n-k})$. Then $\alpha$ and $\beta$ are seen to be injective and surjective, respectively.

Further, $\beta (\alpha (\mathbb{Z_2}^k))=0$ so that $im(\alpha) \subset ker(\beta)$ and $b \in ker(\beta) \implies b\in im(\alpha)$ so that $im(\alpha)=ker(\beta)$. Is this correct?

Also, need help showing how the second is a short exact sequence.