Base of vectors in $\mathbb{Z}_5$

Question

I have little problem. $$((2,4),(1,0))$$ How can I check its base in $\mathbb{ Z}_5^2$ ? Can I just do classic methods like in other set of numbers? Or is it different?

Answer 1

One way is to write it in matrix form: $\begin{pmatrix} 2 & 1\\4 & 0\end{pmatrix}$, viewed as matrix over $\mathbb{Z}_5$.

Then argue that this matrix lies in $\operatorname{GL}_2(\mathbb{Z}_5)$ by showing that its determinant is invertible in $\mathbb{Z}_5$.