Let $e=(a,b,c)$ be a unit vector in $\mathbb{R}^3$ and let $T$ be the linear transformation on $\mathbb{R}^3$ of rotation by $180^\circ$ about $e$. Find the matrix for $T$ with respect to the standard basis $e_1=(1,0,0)$, $e_2=(0,1,0)$ and $e_3=(0,0,1)$.
The rotation matrix in $\mathbb{R}^3$ by $180^\circ$ is : \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} So rotating $e$ by $180^\circ$ gives : \begin{bmatrix} -a \\ -b \\ c \end{bmatrix} After that how to get the transformation matrix w.r.t the standard basis? P.S.- The answer given involves terms consisting of $a,b,c$.