Systems of Linear Equations
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.