Basis and dimension

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$}\right\}.$

True or false? Explain.

1. Every set of $5$ vectors in $\mathbb{R}^{5}$ is linearly independent.

2. Every set of $4$ vectors in $\mathbb{R}^{5}$ is linearly independent.

3. Every set of $6$ vectors in $\mathbb{R}^{5}$ is linearly dependent.

4. No set of $4$ vectors spans $\mathbb{R}^{5}$.

5. Every linearly independent set of $5$ vectors in $\mathbb{R}^{5}$ is a basis of $\mathbb{R}^{5}$.

6. Every linearly independent set of $4$ vectors in $\mathbb{R}^{5}$ is a basis of $\mathbb{R}^{5}$.

7. Some linearly independent set of $4$ vectors i…

If you have $6$ vectors in $\mathbb{R}^{5}$, is it possible they are linearly independent? Explain.