Linear systems

In applied sciences, one is quite often led to face a linear system of the form (5.1)$$Ax = b$$ for the solution of complex problems, where A is a square matrix of dimension n x n whose elements a _ij are either real or complex, while x and b are column vectors of dimension n with x representing the unknown solution and b is a given vector. Component-wise, (5.1) can be written as $$\begin{gathered} {a_{11}}{x_1} + {a_{12}}{x_2} + \ldots + {a_{1n}}{x_n} = {b_1}, \hfill \\ {a_{21}}{x_1} + {a_{22}}{x_2} + \ldots + {a_{2n}}{x_n} = {b_2}, \hfill \\ \vdots \vdots \vdots \hfill \\ {a_{n1}}{x_1} + {a_{n2}}{x_2} + \ldots + {a_{nn}}{x_n} = {b_n}, \hfill \\ \end{gathered} $$.