For the following linear operators $ L$ on $\mathbb{R}^3$, find a matrix $A$ such that $L(x)=Ax$ for every $x$ in $\mathbb{R}^3$

Question

For the following linear operators $L$ on $\mathbb{R}^3$, find a matrix $A$ such that $L(x)=Ax$ for every $x$ in $\mathbb{R}^3$. $L((x_1,x_2,x_3)^T)=(x_3,x_2,x_1)^T$

Answer 1

Since the equation $L(x)=Ax$ is true for all $x$, one way to find $A$ is to simply take $3$ linearly independent $x$ in $\mathbb{R}^3$ and put them in the above equation and get the values of each term in $A$. You can take any linearly independent $x$, but I prefer to use the unit vectors $(0,0,1)^T, (0,1,0)^T, (1,0,0)^T$. After putting each of them in $L(x)=Ax$, you will quickly find that

$$A = \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{matrix} $$

You can also verify this is the matrix by putting a general $y=(y_1,y_2,y_3)^T$ and see that the result is $(y_3,y_2,y_1)^T$