I've been learning linear algebra but can't understand the concepts of linear transformation.
Correct me where i'm wrong:
Say i'm given $T:R^2 \longrightarrow R^2$ is my transformation. This tells me it takes a matrix or vector with two rows, does something to it and returns a matrix with two rows again.
Further it is given the translation occurs over the vector $(-4,2)^T$ and then mirrored in the line $x_2=-x_1$. The transformed matrix must be in homogeneous coordinates.
What do we translate over the vector $(-4,2)^T$ and how does this work?
Your question isn't really clear... Based on what you wrote, my guess is that the linear transformation $T$ is the symmetry relatively to the line of equation $x_2 = -x_1$, and that you want to write its matrix $A$ in the natural base of $\mathbb{R}^2$.
Recall that the two columns of the matrix $A$ are simply the images of the two base vectors $e_1 = (1,0)^T$ and $e_2 = (0,1)^T$ by the transformation $T$.
It is quite obvious (do a drawing to be convinced of it!) that $T(e_1)=-e_2 = (0,-1)^T$ and $T(e_2)=-e_1= (-1,0)^T$, so we have
$$ A = \left(\begin{array}{c c} 0 & -1 \\ -1 & 0 \end{array} \right)$$
And if you want to compute the image of the vector $x = (-4,2)^T$ by the transformation $T$, just compute:
$$ T(x) = Ax =\left(\begin{array}{c c} 0 & -1 \\ -1 & 0 \end{array} \right) \left(\begin{array}{c } -4 \\ 2 \end{array} \right) = \left(\begin{array}{c } -2 \\ 4 \end{array}\right) $$