Exercise 6

Show that each of these is not a vector space. (Hint. Check closure by listing two members of each set and trying some operations on them.)

• Under the operations inherited from $\mathbb{R}^{3}$, this set

  $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\end{pmatrix}\in\mathbb{R}^{3}\mid x+y+z=1\hskip 1.00006pt\}$

• Under the operations inherited from $\mathbb{R}^{3}$, this set

  $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\end{pmatrix}\in\mathbb{R}^{3}\mid x^{2}+y^{2}+z^{2}=1\hskip 1.00006pt\}$

• Under the usual matrix operations,

  $\{\hskip 1.00006pt\begin{pmatrix}a&1\\ b&c\end{pmatrix}\mid a,b,c\in\mathbb{R}\hskip 1.00006pt\}$

• Under the usual polynomial operations,

  $\{\hskip 1.00006pta_{0}+a_{1}x+a_{2}x^{2}\mid a_{0},a_{1},a_{2}\in\mathbb{R}^{+}\hskip 1.00006pt\}$

  where $\mathbb{R}^{+}$ is the set of reals greater than zero

• Under the inherited operations,

  $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\end{pmatrix}\in\mathbb{R}^{2}\mid\text{$x+3y=4$ and $2x-y=3$ and $6x+4y=10$}\hskip 1.00006pt\}$