System of Linear Equations as Continuous Linear Transformation

Theorem
Let $x_1, x_2 \in \R$ be real numbers.

Consider the following system of simultaneous linear equations $\paren S$:

Let $\norm {\, \cdot \,}_2$ be the $2$-norm.

Let $\norm {\, \cdot \,}$ be the supremum operator norm.

Let $I$ be the $2 \times 2$ identity matrix.

Then:


 * $S$ is expressible as $\paren {I - K} x = y$ where $x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$, $y = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$, $K = \begin{pmatrix} \frac 1 2 & \frac 1 3 \\ \frac 1 3 & \frac 1 4 \end{pmatrix}$;


 * $\norm K < 1$;


 * $S$ admits a unique solution.

$\paren S$ is expressible as $\paren {I - K} x = y$
We have that:

Furthermore:

Hence, $\paren S$ can be rewritten as:

Moreover:

By a simple inspection it is evident that $\paren S$ is expressible as $\paren {I - K} x = y$.

$\norm K < 1$
We have that:

Hence:

$S$ admits a unique solution
By Neumann series theorem, $\paren {I - K}^{-1}$ exists in $\map {CL} {\R^2}$.

Hence:

By Inverse of Invertible 2 x 2 Real Square Matrix:

Therefore: