Linear System of Equations | Sahithyan's Notes

A linear system of $m$ equations in $n$ unknowns $x_1, x_2, \ldots, x_n$ is a set of equations of the form

\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 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n &= b_m \end{aligned}

where $a_{ij}, b_i \in \mathbb{R}$ for $1 \le i \le m$ and $1 \le j \le n$ .

In matrix form, $A\vec{x} = \vec{b}$ , where

A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}, \quad \vec{x} = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}, \quad \vec{b} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix}

Here:

$A\vec{x} = \vec{b}$ has exactly one of 3 outcomes:

The system is homogeneous iff $\vec{b} = \vec{0}$ . Otherwise it’s non-homogeneous.

For a system $A\vec{x} = \vec{b}$ , the system $A\vec{x} = \vec{0}$ is its associated homogeneous system.

Methods

Numerical methods for solving linear systems split into 2 classes:

Yield the exact solution in a finite number of steps, in the absence of round-off error.

Useful for special, very large matrices.