Show that the system of equations:
$x_1 = \frac{1}{4}x_1 - \frac{1}{4}x_2 + \frac{2}{15}x_3 +3 $
$x_2 = \frac{1}{4}x_1 + \frac{1}{5}x_2 + \frac{1}{2}x_3 -1 $
$x_3 = -\frac{1}{4}x_1 + \frac{1}{3}x_2 - \frac{1}{3}x_3 +2$
has a unique solution, using the contraction mapping principle.
I approached the question by attempting to choose a clever norm on $\mathbb{R}^3$. Since the sum of the coefficients for each coordinate is less than 1, I chose the norm to be the 1-norm. I'm not really sure where to go from here.