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# Systems of Linear Equations

A system of linear equations with \(m\) equations and \(n\) unknowns \(x_1,\ldots,x_n\) has the form

\[\begin{align*} a_{1,1}\,x_1+\cdots+a_{1,n}\,x_n&=b_1\\ \vdots\\ a_{m,1}\,x_1+\cdots+a_{m,n}\,x_n&=b_m \end{align*}\]

with coefficients \(a_{ij}\) and right-hand sides \(b_1,\ldots,b_m\).

Setting

\[\begin{equation*} A:=\begin{pmatrix}a_{1,1}&\cdots&a_{1,n}\\\vdots&&\vdots\\a_{m,1}&\cdots&a_{m,n}\end{pmatrix}, \qquad x:=\begin{pmatrix}x_1\\\vdots\\x_n\end{pmatrix} \qquad\text{and}\qquad b:=\begin{pmatrix}b_1\\\vdots\\b_m\end{pmatrix} \end{equation*}\]

we may write a system of linear equations in its matrix form

\[\begin{equation*} A\,x=b. \end{equation*}\]

Depending on coeefficient matrix \(A\) and right-hand side \(b\) a system of linear equations either has no solution or exactly one solution or infinitely many solutions.