Solution for equation using contraction mapping

Question

Find a solution of the equation $f(x) = x$ for $x = (x_1, x_2,x_3) \in \mathbb E^3$ (euclidean space) , where $f(x_1, x_2,x_3)= (4x_1+10x_2+5x_3-20, 4x_1+4x_2+3x_3+24, 15x_1+4x_2+15x_3+180)$

How to find a solution for the equation using Banach contraction mapping? the hint from book, it can be solve using matrix form to find the fixed point. Anyone can help me to solve this question ?

Answer 1

Let $x = (1,1,1), y = (0,0,0)$, then we have that $d(x,y) = \sqrt{3} \approx 1.73$. Now, consider $f(x) = (-1,35,214), f(y) = (-20,24,180)$, then we have that $d(f(x),f(y)) \approx 40.4722$. Therefore, $f$ is NOT a contraction mapping. Thus, you should just solve the system of equations $$\begin{align} 4x_1+10x_2+5x_3 - 20 &= x_1\\ 4x_1+4x_2+3x_3 +24 &= x_2 \\ 15x_1+4x_2+15x_3 + 180 &= x_3\end{align}$$ directly. Note, this can be rewritten as $$\begin{align} 3x_1+10x_2+5x_3 &= 20\\ 4x_1+3x_2+3x_3 &= -24 \\ 15x_1+4x_2+14x_3 &= -180\end{align}$$ then we could find the associated (augmented) matrix's reduced row echelon form and the solution can be recovered from that.

EDIT

To beat a dead horse, the reduced form is $$\begin{bmatrix} 1 & 0 & 0 & -4.7273\\ 0 & 1 & 0 & 8.5333\\ 0 & 0 & 1 & -10.2303 \end{bmatrix}$$ hence the final column gives the solution.

Answer 2

$f(x_1,x_2,x_3)=(x_1,x_2,x_3)$ provides three equations in three unknowns, so your usual techniques for solving simultaneous equations will work. The equations are linear, so there will be a single solution. Are you required to use Banach contraction? As presented, it doesn't look like a contraction to me. Some algebra could fix that, but it looks like overkill. Just solve the simultaneous equations.