Working on the book: Robert Messer. "Linear algebra - The gateway to mathematics" (p. 63)
- Write the set $\left\{ a \begin{bmatrix} 2a-b & b+5\\ a+2 & a+b \end{bmatrix} \in \mathbb{M}(2,2) \Bigm| a,b\in \mathbb{R}\right\}$ as a plane in $\mathbb{M}(2,2)$.
The solution is:
$$\left\{ a \begin{bmatrix} 2 & 0\\ 1 & 1 \end{bmatrix}+ b \begin{bmatrix} -1 & 1\\ 0 & 1 \end{bmatrix}+ \begin{bmatrix} 0 & 5\\ 2 & 0 \end{bmatrix} \Bigm| a,b\in \mathbb{R}\right\}$$
- Does this representation has the form $\left\{ r\mathbf{v}+s\mathbf{w}+\mathbf{x} : r,s \in \mathbb{R}\right\}$ ? If so, do $\mathbf{v}$, $\mathbf{w}$ and $\mathbf{x}$ consist of two pairs of vectors in $\mathbb{R}^2$.
- Which are the direction vectors in this set ?
I cannot see how the solution set represents a plane and which are the vectors (I see three matrices there).
Use the basis $e_1=\begin{pmatrix}1 & 0\\ 0 & 0\end{pmatrix}$, $e_2=\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix}$, $e_3=\begin{pmatrix}0 & 0\\ 1 & 0\end{pmatrix}$, $e_4=\begin{pmatrix}0 & 0\\ 0 & 1\end{pmatrix}$. Do you recognize your matrices as vectors now?