## Quiz 5: Example and Non-Example of Subspaces in 3-Dimensional Space

## Problem 304

**Problem 1** Let $W$ be the subset of the $3$-dimensional vector space $\R^3$ defined by

\[W=\left\{ \mathbf{x}=\begin{bmatrix}

x_1 \\

x_2 \\

x_3

\end{bmatrix}\in \R^3 \quad \middle| \quad 2x_1x_2=x_3 \right\}.\]

** (a)** Which of the following vectors are in the subset $W$? Choose all vectors that belong to $W$.

\[(1) \begin{bmatrix}

0 \\

0 \\

0

\end{bmatrix} \qquad(2) \begin{bmatrix}

1 \\

2 \\

2

\end{bmatrix} \qquad(3)\begin{bmatrix}

3 \\

0 \\

0

\end{bmatrix} \qquad(4) \begin{bmatrix}

0 \\

0

\end{bmatrix} \qquad(5) \begin{bmatrix}

1 & 2 & 4 \\

1 &2 &4

\end{bmatrix} \qquad(6) \begin{bmatrix}

1 \\

-1 \\

-2

\end{bmatrix}.\]

**(b)** Determine whether $W$ is a subspace of $\R^3$ or not.

**Problem 2** Let $W$ be the subset of $\R^3$ defined by

\[W=\left\{ \mathbf{x}=\begin{bmatrix}

x_1 \\

x_2 \\

x_3

\end{bmatrix} \in \R^3 \quad \middle| \quad x_1=3x_2 \text{ and } x_3=0 \right\}.\]
Determine whether the subset $W$ is a subspace of $\R^3$ or not.