Subspaces of R^n

Which of the following sets are subspaces of $\mathbb{R}^{3}$? Explain.

1. $V_{1}=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]~{}\right|~{}\left|u_{1}\right|\leq 4\right\}$.

2. $V_{2}=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]~{}\right|~{}\text{$u_{i}\geq 0$ for each $i=1,2,3$}\right\}$.

3. $V_{3}=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]~{}\right|~{}u_{3}+u_{1}=2u_{2}\right\}$.

4. $V_{4}=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]~{}\right|~{}u_{3}\geq u_{1}\right\}$.

5. $V_{5}=\left\{\left.\mathbf{u}=\left[\begin{array}{c}u_{1}\\ u_{2}\\ u_{3}\end{array}\right]~{}\right|~{}u_{3}=u_{1}=0\right\}$.

Let $\mathbf{w}\in\mathbb{R}^{4}$ be a given fixed vector. Let

Is $M$ a subspace of $\mathbb{R}^{4}$? Explain.

Let $\mathbf{w},\mathbf{v}$ be given vectors in $\mathbb{R}^{4}$ and define

Is $M$ a subspace of $\mathbb{R}^{4}$? Explain.

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

Suppose $V,W$ are subspaces of $\mathbb{R}^{n}$. Let $V\cap W$ be the set of all vectors that are in both $V$ and $W$. Show that $V\cap W$ is also a subspace.