Let $T(n;\mathbb{R}) \subset M(n;\mathbb{R})$ denote the set of all matrices whose trace is zero. Write down a basis for $T(2;\mathbb{R})$. What is the quotient space $M(n;\mathbb{R})/T(n;\mathbb{R})$ isomorphic to?
Answer of the first part is $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $\begin{pmatrix} 0 & 0 \\ 1 & -0 \end{pmatrix}$ but i am not sure about second part. My guess is it is $\mathbb{R}-\{0\}$.