Subsets and Subspaces

Refer to Vocabulary for definitions. A subspace MUST include the zero vector.

Set Builder Notation

Say you want to create a set with matrixes in ${\mathbb{R}}^2$ such that the top element $\times$ the bottom element $=0$. Writing out each element is not only impractical, but impossible. Hence we use set builder notation,

\left\{ \text{matrix in } \mathbb{R}^2 \text{ such that the top element }\times \text{ the bottom element } = 0 \right\}$$

\left{ \text{matrix} \in \mathbb{R}^2 \text{ } | \text{ } \text{ the top element }\times \text{ the bottom element } = 0 \right}$$

$$\boxed{\left\{\left[\begin{array}{c}a\\ b\end{array}\right]\in {\mathbb{R}}^2|a\times b=0\right\}}$$

Is $v=\left\{\left[\begin{array}{c}a\\ b\end{array}\right]\in {\mathbb{R}}^2|a\times b=0\right\}$ a subspace?

$\left[\begin{array}{c}0\\ 1\end{array}\right],\left[\begin{array}{c}1\\ 0\end{array}\right]\in v$ but $\left[\begin{array}{c}0\\ 1\end{array}\right]+\left[\begin{array}{c}1\\ 0\end{array}\right]\notin v$ So $v$ is not a subspace.