Exercise 17

At this point “the same” is only an intuition, but nonetheless for each vector space identify the $k$ for which the space is “the same” as $\mathbb{R}^{k}$.

• The $2\!\times\!3$ matrices under the usual operations

• The $n\!\times\!m$ matrices (under their usual operations)

• This set of $2\!\times\!2$ matrices

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

• This set of $2\!\times\!2$ matrices

  $\{\hskip 1.00006pt\begin{pmatrix}a&0\\ b&c\end{pmatrix}\mid a+b+c=0\hskip 1.00006pt\}$