# Systems of linear equations

A system of linear equations has the form:

\begin{align}
a_{11}x_{1} + \cdots + a_{1n}x_{n} &= b_{1} \\
\vdots \\
a_{m1}x_{1} + \cdots + a_{mn}x_{n} &= b_{n}
\end{align}

There are *n* unknown variables, $(x_1, \cdots, x_n)$ and *m* equations. The
*m* equations have coefficients $a_{mn}$ for each of the *n* variables and
*m* right-hand side values $(b_1, \cdots, b_m)$. These equations be rewritten
using matrices and vectors:

\begin{equation}
\begin{bmatrix}
a_{11} & \cdots & a_{1n} \\
\vdots &     & \vdots \\
a_{m1} & \cdots & a_{mn}
\end{bmatrix} \begin{bmatrix} \ x_{1} \\ \vdots \\ x_{n} \end{bmatrix} =
\begin{bmatrix} b_{1} \\ \vdots \\ b_{m} \end{bmatrix}
\end{equation}

or in shorthand:

\begin{equation}
\vv{A}\vv{x} = \vv{b}
\end{equation}

A system of linear equations can have:

- No solution (inconsistent equations)
- One solution
- Infinitely many solutions (undetermined)

For example, consider the following graphs of different systems of 2 equations:

- $x+y = 1$, $x+y = 0$

  ```{image} ./_images/nosolution.jpg
  :alt: One Solution
  :width: 300px
  :align: center
  ```

  This system of equations has no solution because the two lines are parallel
  and never intersect.

- $x + y = 1$, $2x - y = 0$

  ```{image} ./_images/onesolution.jpg
  :alt: One Solution
  :width: 300px
  :align: center
  ```

  This system of equations has one solution because the two lines intersect at a
  point.

- $x+y = 1$, $2x + 2y = 2$

  ```{image} ./_images/infinite.jpeg
  :alt: One Solution
  :width: 300px
  :align: center
  ```

   This system of equations has infinitely many solutions because the two lines
   are parallel and completely overlap.

```{topic} Types of systems
For a system of linear equations $\vv{A}\vv{x} = \vv{b}$, if the matrix **A**
is *m* x *n*, then we say that the system is:

- *Undetermined* if $m < n$ (fewer equations than unknowns). The system has no
  unique solution.
- *Determined* if $m = n$ (same number of equations and unknowns).
- *Overdetermined*: $m > n$ (more equations than unknowns)
```