Let $G$ be the linear operator on $R^3$ defined by $G(x,y,z)=(2y+z,x-4y,3x).$

$($a$)$ Find the matrix representation of $G$ relative to the basis

$S=\{w_1,w_2,w_3\}=\{(1,1,1),(1,1,0),(1,0,0)\}$

$($b$)$ Verify that $[G][v]=[G(v)]$ for any vector $v$ in $R^3.$