Let $L:\mathbb{R}^3 \to \mathbb{R}^3$ be the linear operator defined by

$$L(x) = \left[\begin{array}{c} 6 x_1 + 12 x_2 - 6 x_3\\ 6 x_2\\ 18 x_1 - 18 x_3 \end{array}\right]$$

(a) Find the dimension of the range of $L$:

(b) Find the dimension of the kernel of $L$:

(c) Let $S$ be the subspace of $\mathbb{R}^3$ spanned by $24 e_1$ and $7 e_2 + 7 e_3$. Find the dimension of $L(S)$: