A linear equation for unknowns $x_1,\dots ,x_n$ is of form

i.e., a linear combination of $x_1,\dots ,x_n$ is set equal to some value ($b_1$)

Multiple ($m$) linear equations form a linear system

The above is concisely stated as $𝑨𝒙=𝒃$, $𝑨\in {ℝ}^{m\times n}$, $𝒙\in {ℝ}^n$, $𝒃\in {ℝ}^m$

Linear systems can have:

• No solutions: $0\cdot x=1$; $x+y=1$, $x+y=2$

• Unique solution: $x=1$; $x+y=1$, $x-y=0$

• Infinitely many solutions: $x+y=1$, $2x+2y=2$.

Some systems $𝑨𝒙=𝒃$, $𝑨\in {ℝ}^{m\times m}$ are easily solvable:

• Diagonal systems $𝑨=\mathrm{diag}\left(\left[\begin{array}{llll}{a}_{11}& a_{22}& \dots & a_{mm}\end{array}\right]\right)$. If $a_{ii}\ne 0$ for all $i$ then

  ${\displaystyle x_i=b_i/a_{ii}}$

• Triangular systems, $𝑨$ is upper triangular if all elements beneath diagonal are zero, solvable by back-substitution

  ${\displaystyle 𝑨=\left[\begin{array}{lllll}{a}_{11}& a_{12}& a_{13}& \dots & a_{1m}\\ 0& a_{22}& a_{23}& \dots & a_{2m}\\ 0& 0& a_{33}& \dots & a_{3m}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & a_{mm}\end{array}\right]}$

  Example:

  ${\displaystyle \{\begin{array}{l}x+3y-2z=5\\ 2y-6z=4\\ 3z=6\end{array}.\Rightarrow z=2,y=8,x=-15.}$