Linear equations¶

System of linear equations

Let \(a_{i,j},b_i\in F\) be constants where \(i=1,2,\dots,m,j=1,2,\dots,n\). The equations of the form

\[ \begin{cases} b_1&=a_{1,1}x_1+a_{1,2}x_2+\dots+a_{1,n}x_n\\ b_2&=a_{2,1}x_1+a_{2,2}x_2+\dots+a_{2,n}x_n\\ \vdots&\\ b_m&=a_{m,1}x_1+a_{m,2}x_2+\dots+a_{m,n}x_n\\ \end{cases} \]

is called a system of linear equations.

It is clear that we can write it as \(Ax=b\) where \(x,b\in F^m,A\in\text{M}_{m\times n}(F)\) and \(A_{i,j}=a_{i,j}\).

Consistency

A system of linear equations \(Ax=b\) is said to be consistent if there is \(x_0\in F^m\) such that \(Ax_0=b\). It is said to be inconsistent if it is not consistent.

Augmented matrix

Let \(A\in\text{M}_{m\times n}(F),B\in\text{M}_{m\times p}(F)\) where \(F\) is a field. The augmented matrix of \(A\) and \(B\), denoted by \([\,A\,|\,B\,]\) is a \(m\times (n+p)\) matrix that

\[ [\,A\,|\,B\,]_{i,j}=\begin{cases}A_{i,j},&\text{if }j\le n\\B_{i,j},&\text{if }j>n\end{cases} \]

We also say that \(A\) is augmented with \(B\) or \(B\) augments with \(A\) in this case.

Gaussian elemination