Exercise 5.4.11

In this exercise, we use scalars from the field $\mathbb{Z}_{5}$ of integers modulo $5$ instead of real numbers (see Section an exercise given elsewhere in the book, “Fields”). Find a basis and the dimension of each of the following subspaces of $(\mathbb{Z}_{5})^{n}$.

1. $V_{1}=\operatorname{span}\left\{\left[\begin{array}{r}2\\ 3\\ 1\\ 4\end{array}\right],~{}\left[\begin{array}{r}1\\ 4\\ 3\\ 2\end{array}\right],~{}\left[\begin{array}{r}4\\ 1\\ 2\\ 3\end{array}\right],~{}\left[\begin{array}{r}2\\ 2\\ 1\\ 0\end{array}\right]\right\}$.

2. $V_{2}=\left\{\left.\left[\begin{array}{c}u\\ v\\ w\end{array}\right]~{}\right|~{}\text{$2u+v=0$ and $u+4w=0$}\right\}.$

3. $V_{3}=\left\{\left.\left[\begin{array}{c}u\\ v\\ w\end{array}\right]~{}\right|~{}\text{$u+2v+3w=0$}\right\}.$