Let $T: \mathbb{R}^2 \to \mathbb{R}^3$ be a linear map given by
$T((x, y)^T) = (x + y, 2x + 3y, 3x + 4y)^T$.
(a) Find the matrix representation of $T$ with respect to the standard bases of $\mathbb{R}^2$ and $\mathbb{R}^3$.
(b) Find the kernel and the image of $T$.
(c) Find nullity($T$) and rank($T$).