Exercise 1

Which of these subsets of the vector space of $2\!\times\!2$ matrices are subspaces under the inherited operations? For each one that is a subspace, parametrize its description. For each that is not, give a condition that fails.

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

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

• $\{\hskip 1.00006pt\begin{pmatrix}a&0\\ 0&b\end{pmatrix}\mid a+b=5\hskip 1.00006pt\}$

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