2. The matrix formulation

You know how to write a square system of linear equations with matrices; an overdetermined system can be written the same way. In our example:

$\left[\begin{array}{cc} 1&2\\ 2&1\\ 1&-2\\ 2&-1 \end{array}\right]\left[\be... ...ray}\right] \approx \left[\begin{array}{c} 3\\ 3\\ 1\\ 2 \end{array}\right]$. In general, an overdetermined system is of the form $A$   x$\approx$   b.

Here $A$ is $m \times n$ with $m > n$, x is an $n$-vector, and b is an $m$-vector. The errors can also be expressed as a single vector $\epsilon = \langle \epsilon _ 1,\dots, \epsilon _ m \rangle$. If $\epsilon$ is written as a column vector, then $\epsilon = A$   x$-$   b.