Matrices and Linear Systems

What is a linear system?

A linear system is a set of linear equations, or equations of the following form, where $a_n$ and $c$ are constants, and $x_n$ are variables.

$$a_1x_1 + a_2x_2 + a_3x_3 + \ldots + a_nx_n = c$$

An example of a linear system:

$$ \begin{array}{rcrcrcr} 3x_1 &+& 12x_2 &-& 3x_3 &=& 2\\ 4x_1 & & &+& 2x_3 &=& -1\\ 12x_1 &-& x_2 & & &=& 0 \end{array} $$

This is often represented in the following form, called an augmented matrix. On the left are each of the coefficients from the system, and on the right are each of the constants.

$$ \left[\begin{array}{rrr|r} 3 & 12 & -3 & 2\\ 4 & 0 & 2 & -1\\ 12 & -1 & 0 & 0 \end{array}\right] $$

Anatomy of a matrix

Each number in a matrix is called an entry. For example, in the matrix above, $3$ and $2$ are examples of entries. A horizontal set of entries is called a row, and a vertical set is called a column. When an augmented matrix is used to represent a system of linear equations, the last column is called the constant column and the other columns are called variable columns. Each row respresents one linear equation in the system.

Exercises

Here are some practice problems for translating a system to an augmented matrix.