From systems to matrices

Systems of linear equations and matrices: Systems and matrices

From systems to matrices

The elimination method, which solves systems of linear equations by use of 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 is often framed in round brackets. Since the unknowns \(x_1, \ldots, x_n\) and 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; "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: \[A=\begin{pmatrix}a_{11} & a_{12} & \cdots & a_{1n} \\
\vdots & \vdots & & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}\end{pmatrix}\quad\mathrm{and}\quad \vec{b}=\cv{b_1 \cr \vdots\cr b_m\cr}\]

With this notation and with # \vec {x} = \begin{pmatrix}x_1\\ \vdots \\ x_n\end{pmatrix} #, we can briefly describe the above system as \[A\vec{x} = \vec{b}\]

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 mostly 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 for describing the system.

Later 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 written 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 yet another form in which a system of linear equations can be represented: the vector form. Here, too, each unknown occurs at most once, namely as a scalar coefficient of a vector that corresponds to a column of the coefficient matrix.

Give the homogeneous system of linear equations in the unknowns \(x,y,z\) (in this order) that has the following coefficient matrix: \[\left(\begin{array}{rrr}1 & -1 & -1 \\ 0 & -1 & -1 \\ 1 & -2 & -1 \end{array}\right) \] You can enter the system as three linear equations that are linked by the logic and: #\land#.

It is also possible to click on the following expression, and to replace the three little boxes by the appropriate equations:
\[\lineqs{&\Box& \\ &\Box& \\ &\Box & \\ }\]
Do not carry out any simplifications, but make the direct translation from the matrix to the system.

System: \[\lineqs{ x-y-z&=&0\cr -y-z&=&0\cr x-2 y-z&=&0 }\]

This system of linear equations entered in at least two ways:

by using the logic and, #\land#, which can be entered by clicking its icon on the virtual palette under the tab logic.
by clicking on the left brace on the virtual palette under the tab vector; it is set for two rows, but you can (in due course *** ***) you can add a row by pressing the down arrow on your keyboard.

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