Linear Systems

Abstract

In this chapter we deal with the numerical solution of a system of n linear equations

$$ \begin{gathered} {a_{{11}}}{x_1} + {a_{{12}}}{x_2} + \cdots + {a_{{1n}}}{x_n} = {b_1} \hfill \\ {a_{{21}}}{x_1} + {a_{{22}}}{x_2} + \cdots + {a_{{2n}}}{x_n} = {b_2} \hfill \\ \vdots \quad \quad \quad \quad \quad \quad \quad \quad \vdots \quad \quad \, \vdots \hfill \\ {a_{{n1}}}{x_1} + {a_{{n2}}}{x_2} + \cdots + {a_{{nn}}}{x_n} = {b_n} \hfill \\ \end{gathered} $$

or, in short notation,

$$ Ax = b $$

where A ∈ Mat _n (R) is a real (n, n)-matrix and b, x ∈ Rⁿ are real n-vectors. Before starting to compute the solution x, we should ask ourselves: When is a linear equation system solvable at all?