Exercise 5.3.4

In this exercise, we use scalars from the field $\mathbb{Z}_{2}$ of integers modulo $2$ instead of real numbers (see Section an exercise given elsewhere in the book, “Fields”). Which of the following sets are subspaces of $(\mathbb{Z}_{2})^{3}$?

1. $V_{1}=\left\{\left[\begin{array}{r}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{r}0\\ 1\\ 1\end{array}\right],\left[\begin{array}{r}1\\ 0\\ 1\end{array}\right],\left[\begin{array}{r}0\\ 0\\ 0\end{array}\right]\right\}$.

2. $V_{2}=\left\{\left[\begin{array}{r}1\\ 1\\ 1\end{array}\right],\left[\begin{array}{r}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{r}1\\ 0\\ 0\end{array}\right],\left[\begin{array}{r}0\\ 0\\ 0\end{array}\right]\right\}$.

3. $V_{3}=\left\{\left[\begin{array}{r}0\\ 0\\ 0\end{array}\right]\right\}$.

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