From systems of linear equations to matrices

Systems of linear equations: System of linear equations and matrices

From systems of linear equations to matrices

The elimination method, in which systems of linear equations are solved with elementary operations, actually works only with the coefficients and constants of the system. A good accounting in the form of a succinct notation can help expedite this process.

The system of \(m\) linear equations with \(n\) unknowns \(x_1, \ldots, x_n\) \[\left\{\;\begin{array}{llllllllll} a_{11}x_1 \!\!\!\!&+&\!\!\!\! a_{12}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{1n}x_n\!\!\!\!&=&\!\!\!\!b_1\\ a_{21}x_1 \!\!\!\!&+&\!\!\!\! a_{22}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{2n}x_n\!\!\!\!&=&\!\!\!\!b_2\\ \;\;\vdots && \vdots &&&& \vdots&&\!\!\!\!\vdots\\ a_{m1}x_1 \!\!\!\!&+&\!\!\!\! a_{m2}x_2 \!\!\!\!&+&\!\!\!\! \cdots \!\!\!\!&+&\!\!\!\! a_{mn}x_n\!\!\!\!&=&\!\!\!\!b_m\end{array}\right.\] is often written as follows:

\[\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}\!\!\begin{pmatrix}x_1\\ \vdots \\ x_n\end{pmatrix}= \begin{pmatrix}b_1\\ \vdots \\ b_m\end{pmatrix}\]

Such a rectangular array is called a matrix, and it is often framed in round brackets. Because the unknowns \(x_1, \ldots, x_n\) their order of appearance does not change during the solving process, the system is also well represented by the matrix \[\left(\,\begin{array}{cccc|c}
a_{11} & a_{12} & \cdots & a_{1n} & b_1\\ a_{21} & a_{22} & \cdots & a_{2n} & b_2\\ \vdots & \vdots & & \vdots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m\end{array}\,\right)\] The vertical line is drawn to indicate that the last column represents the right-hand side of the system. In general, however, this line is omitted. This matrix is called the augmented matrix of the system; "augmented" because the matrix can be viewed as \[(A|\vec{b})\] where \(A\) stands for the coefficient matrix of the system and \(\vec{b}\) for the column of numbers at the right-hand sides of the equations: \[A=\begin{pmatrix}a_{11} & a_{12} & \cdots & a_{1n} \\
\vdots & \vdots & & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}\end{pmatrix}\quad\mathrm{en}\quad \vec{b}=\cv{b_1 \cr \vdots\cr b_m\cr}\]

With this notation and with #\vec{x} =\begin{pmatrix}x_1\\ \vdots \\ x_n\end{pmatrix}# we can briefly describe the above system as \[A\vec{x} = \vec{b}\] We speak of the matrix form of the system of linear equations or call it simply the corresponding matrix equation.

A systematic way of describing the system of equations is:

\[\begin{array}{|cccc|c|} \hline
{x_{1}} & x_{2} & \cdots & x_{n} & \\ \hline
{a_{11}} & a_{12} & \cdots & a_{1n} & b_1\\
a_{21} & a_{22} & \cdots & a_{2n} & b_2\\
\vdots & \vdots & & \vdots & \vdots \\
a_{m1} & a_{m2} &\cdots & a_{mn} & b_m \\ \hline \end{array}\]
The names of the unknowns are in most case known, so we can actually omit the top row: \[\begin{array}{|cccc|c|} \hline
{a_{11}} & a_{12} & \cdots & a_{1n} & b_1\\
a_{21} & a_{22} & \cdots & a_{2n} & b_2\\
\vdots & \vdots & & \vdots & \vdots \\
a_{m1} & a_{m2} &\cdots & a_{mn} & b_m \\ \hline \end{array}\] What remains is a rectangular array of numbers for the coefficients and right-hand sides of the system of linear equations.

Homogeneous

If the column vector \(\vec{b}\) consists only of zeros, then the system is homogeneous. In this case, the coefficient matrix suffices to describe the system.

In the next chapter we will introduce the product of a matrix by a column vector. Then it will become clear that the given system of linear equations can be witten in matrix notation as \[\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}\cdot\begin{pmatrix}x_1\\ \vdots \\ x_n\end{pmatrix}= \begin{pmatrix}b_1\\ \vdots \\ b_m\end{pmatrix}\] where #\cdot# indicates the multiplication. In shorthand, this then becomes \[A\cdot \vec{x} =\vec{b}\] Usually we omit the symbol #\cdot# for multiplication and write \[\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix}\!\!\begin{pmatrix}x_1\\ \vdots \\ x_n\end{pmatrix}= \begin{pmatrix}b_1\\ \vdots \\ b_m\end{pmatrix}\] and \[A\vec{x} =\vec{b}\]

Vector form

Earlier we discussed some other form to a system of linear equations, the vector shape. Here, too, each unknown is no more than one time before and scalar or as a vector corresponding to a column of the coefficient matrix.