System of linear equations

Vectors and matrices play an important role in describing and solving systems of linear equations. Let a linear system of \(m\) equations in \(n\) unknown parameters \(x_{i}\), \(i=1, \ldots, n\), be given as

\[\begin{array}{ccc} y_{1} & = & a_{11}x_{1}+a_{12}x_{2}+\ldots + a_{1n}x_{n} \\ y_{2} & = & a_{21}x_{1}+a_{22}x_{2}+\ldots + a_{2n}x_{n} \\ \vdots & & \vdots \\ y_{m} & = & a_{m1}x_{1}+a_{m2}x_{2}+\ldots + a_{mn}x_{n} \end{array} \]

This system of linear equations can be written in a matrix form as:

\[ \begin{bmatrix} y_1\\ y_2 \\ \vdots \\ y_m \end{bmatrix}= \begin{bmatrix} a_{11}&a_{12}&\dots&a_{1n}\\a_{21}&a_{22}&\dots&a_{2n} \\ \vdots&\vdots&\vdots&\vdots \\ a_{m1}&a_{m2}&\dots&a_{mn} \end{bmatrix} \begin{bmatrix} x_1\\ x_2 \\ \vdots \\ x_n \end{bmatrix}\]

By introducing  \[y=\begin{bmatrix} y_1\\ y_2 \\ \vdots \\ y_m \end{bmatrix},  ~  A=\begin{bmatrix} a_{11}&a_{12}&\dots&a_{1n}\\a_{21}&a_{22}&\dots&a_{2n} \\ \vdots&\vdots&\vdots&\vdots \\ a_{m1}&a_{m2}&\dots&a_{mn} \end{bmatrix}, ~ \text{and} ~ x=\begin{bmatrix} x_1\\ x_2 \\ \vdots \\ x_n \end{bmatrix},\] the linear system can be written in the compact matrix form \(y = A x\).

Note that the right-hand side of the system \(y = A x\) can be re-written as the linear combinations of the column vectors of A:

\[Ax=\begin{bmatrix} a_{11}\\a_{21}\\ \vdots \\ a_{m1} \end{bmatrix}x_1+\begin{bmatrix} a_{12}\\a_{22}\\ \vdots \\ a_{m2} \end{bmatrix}x_2 + \dots +\begin{bmatrix} a_{1n}\\a_{2n}\\ \vdots \\ a_{mn} \end{bmatrix}x_n.\]

The linear system of equations is solved, once it is known which linear combination(s) of the columns of \(A\) produces \(y\).

Example: Assume the following system of three equations with two unknowns:

\[1=x_1-x_2\] \[-1=x_1-3x_2\] \[3=x_1+x_2\]

These three equations can be written in a matrix form as

\[\begin{bmatrix} 1\\-1\\3 \end{bmatrix} = \begin{bmatrix} 1&-1\\1&-3\\1&1 \end{bmatrix} \begin{bmatrix} x_1\\x_2 \end{bmatrix}\]

The solution of this system is \(x_1=2\) and \(x_2=1\). We can see that the particular linear combination of columns of \(A\) using \(x_1=2\) and \(x_2=1\) produces the vector \(\begin{bmatrix} 1\\-1\\3 \end{bmatrix}\) :  

\[\begin{bmatrix} 1\\1\\1 \end{bmatrix}2 + \begin{bmatrix} -1\\-3\\1 \end{bmatrix}1=\begin{bmatrix} 1\\-1\\3 \end{bmatrix}.\]