Linear Algebra – Systems of linear equations

A linear equation in de variables \(x_1,x_2,\ldots,x_n\) is an equation that can be written in the form

\[a_1x_1+a_2x_2+\cdots+a_nx_n=b,\]

where \(b\) and the coefficients \(a_1,a_2,\ldots,a_n\) are real or coomplex numbers and \(n\) is a positive integer.

A system of linear equations is a collection of one or more linear equations in the same variables, say \(x_1,x_2,\ldots,x_n\). Sometimes this is called a linear system for short, however this is not really correct (the equations are linear, not the system).

A solution of the system is a list \(\{s_1,s_2,\ldots,s_n\}\) of numbers that makes each equation a true statement when the values \(s_1,s_2,\ldots,s_n\) are substituted for \(x_1,x_2,\ldots,x_n\), respectively.

The set of all possible is called the solution set of the linear system. Two linear systems are called equivalent if they have the same solution set.

For a system of linear equations there are three possibilities: the system has

1. no solutions,


2. exactly one solution,


3. infinitely many solutions.

A system of linear equations is called consistent if it has either one solution or infinitely many solutions and it is called inconsistent if it has no solutions.

The essential information of a system of linear equations can be displayed in an augmented matrix:

\[\left\{\begin{array}{cccccccc}a_{11}x_1&+&a_{12}x_2&+&\cdots&+&x_{1n}x_n&=&b_1\\[2.5mm] a_{21}x_1&+&a_{22}x_2&+&\cdots&+&x_{2n}x_n&=&b_2\\[2.5mm] \vdots&&\vdots&&&&\vdots&&\vdots\\[2.5mm] a_{m1}x_1&+&a_{m2}x_2&+&\cdots&+&a_{mn}x_n&=&b_m\end{array}\right.\quad\Longrightarrow\quad \left(\left.\begin{array}{cccc}a_{11}&a_{12}&\ldots&a_{1n}\\[2.5mm] a_{21}&a_{22}&\ldots&a_{2n}\\[2.5mm] \vdots&\vdots&&\vdots\\[2.5mm] a_{m1}&a_{m2}&\ldots&a_{mn}\end{array}\;\right|\; \begin{array}{c}b_1\\[2.5mm]b_2\\[2.5mm]\vdots\\[2.5mm]b_m\end{array}\right).\]

Here the matrix \(\begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\[2.5mm] a_{21}&a_{22}&\ldots&a_{2n}\\[2.5mm] \vdots&\vdots&&\vdots\\[2.5mm] a_{m1}&a_{m2}&\ldots&a_{mn}\end{pmatrix}\) is called the coefficient matrix of the system of linear equations.

The size of such a matrix tells how many rows and columns it has; this is an \(m\times n\) matrix with \(m\) rows and \(n\) columns.

The Gauss elimination (the row reduction proces).

There are three elementary row operations:

1. Replacement\({}^{(*)}\): replace a row by the sum of that row and a multiple of another row,


2. Interchange: interchange two rows,


3. Scaling: multiply all entries in a row by a nonzero constant.

\({}^{(*)}\) usually one says: "add to one row a multiple of another row".

Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into the other. Note that all elementary row operations are reversable.

If the augmented matrices of two systems of linear equations are row equivalent, then the two systems have the same solution set.

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Last modified on March 22, 2021