Exercise 12

For each, decide if it is a vector space; the intended operations are the natural ones.

• The diagonal $2\!\times\!2$ matrices

  $\{\hskip 1.00006pt\begin{pmatrix}a&0\\ 0&b\end{pmatrix}\mid a,b\in\mathbb{R}\hskip 1.00006pt\}$

• This set of $2\!\times\!2$ matrices

  $\{\hskip 1.00006pt\begin{pmatrix}x&x+y\\ x+y&y\end{pmatrix}\mid x,y\in\mathbb{R}\hskip 1.00006pt\}$

• This set

  $\{\hskip 1.00006pt\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}\in\mathbb{R}^{4}\mid x+y+z+w=1\hskip 1.00006pt\}$

• The set of functions $\{\hskip 1.00006pt\text{$f\colon\mathbb{R}\to\mathbb{R}$}\mid df/dx+2f=0\hskip 1.00006pt\}$

• The set of functions $\{\hskip 1.00006pt\text{$f\colon\mathbb{R}\to\mathbb{R}$}\mid df/dx+2f=1\hskip 1.00006pt\}$