2: Systems of Linear Equations
Consider the system of \(n\) linear equations and \(n\) unknowns, given by
\[\begin{aligned}a_{11}x_1+a_{12}x_2+\cdots +a_{1n}x_n&=b_1, \\ a_{21}x_1+a_{22}x_2+\cdots +a_{2n}x_n&=b_2, \\ \qquad \vdots\qquad\qquad &\vdots \\ a_{n1}x_1+a_{n2}x_2+\cdots +a_{nn}x_n&=b_n.\end{aligned} \nonumber \]
We can write this system as the matrix equation
\[\text{Ax}=\text{b},\label{eq:1} \]
with
\[\text{A}=\left(\begin{array}{cccc}a_{11}&a_{12}&\cdots &a_{1n} \\ a_{21}&a_{22}&\cdots &a_{2n} \\ \vdots&\vdots&\ddots&\vdots \\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{array}\right),\quad\text{x}=\left(\begin{array}{c}x_1\\x_2\\ \vdots\\x_n\end{array}\right),\quad\text{b}=\left(\begin{array}{c}b_1\\b_2\\ \vdots\\ b_n\end{array}\right).\nonumber \]
This chapter details the standard algorithm to solve \(\eqref{eq:1}\).