Solving systems of linear equations is the foundational problem of linear algebra.
Section 2.1 Systems of Linear Equations
In high-school algebra, we consider systems of two linear equations in two unknowns. For example,
\begin{align*}
x + 2y \amp = 5\\
2x + 3y \amp = 8
\end{align*}
These systems can be solved graphically, using substitution, or using elimination.
In linear algebra, we develop more advanced and general techniques to solve larger systems of equations.
Subsection 2.1.1 Motivation: Systems of Two Linear Equations In Two Variables
A system of two linear equations in two variables is of the form,
\begin{align*}
a_{11} x_1 + a_{12} x_2 \amp = b_1 \\
a_{21} x_1 + a_{22} x_2 \amp = b_2
\end{align*}
Graphically, these two linear equations represent lines in the plane, say denoted by \(L_1\) and \(L_2\text{,}\) respectively. Then, an ordered pair \((s_1,s_2)\) is a solution to the system if it satisfies both equations in the system. Graphically, this means that the point \((s_1,s_2)\) lies on both lines \(L_1\) and \(L_2\text{,}\) i.e. is a point of intersection.
To solve a system is to determine all of its soltions. The set of all solutions of a system is called its solution set.
In general, there are 3 cases for the intersection of two lines in the plane:
- The lines can intersect at a single point.
- The lines can be parallel to each other, and so never intersect.
- The lines can coincide (overlap each other), and so intersect at infinitely many points.
Graphing both lines to solve the system is sometimes called the row picture, because it considers each equation (each “row”) of the system. This is in contrast to the column picture, which will be considered later on.
In the language of solutions of linear systems, these correspond to the possibilities of the system having,
- Exactly one solution.
- No solution.
- Infinitely many solutions.
Definition 2.1.1.
A system of linear equations is consistent if it has at least one solution. Otherwise, it is inconsistent, i.e. if it has no solution.Subsection 2.1.2 Introduction to Systems of Linear Equations
Recall that in algebra, the degree of a term is the sum of the powers of the variables it contains. For example, \(5x^2\) has degree 2, \(4xyz\) has degree 3, and \(x\) has degree 1. A linear equation, broadly, is an equation where every term has degree at most 1. Here, we define a general linear equation in \(n\) variables.
Definition 2.1.2.
A linear equation in the variables \(x_1, \dots, x_n\) is an equation which can be written in the form,
\begin{equation*}
a_1 x_1 + a_2 x_2 + \dots + a_n x_n = b
\end{equation*}
where \(a_1, \dots, a_n, b \in \mathbb{R}\) are called the coefficients of the equation.
Most considered examples will contain somewhere from 2-5 variables (i.e. \(n = 2, \dots, 5\)). However, most modern real-life applications of linear algebra contain hundreds or thousands of variables.
Note 2.1.3.
The variables are denoted by \(x_1, \dots, x_n\) rather than \(x, y, z\) etc. in order to conveniently represent a large number of equations. If there are only two or three equations, then \(x, y\text{,}\) and \(z\) are sometimes used.Definition 2.1.4.
A system of linear equations, broadly, is a collection of one or more linear equations involving the same variables. More precisely, a system of \(m\) linear equations in \(n\) variables is of the form,
\begin{alignat*}{3}
a_{11}x_1 \amp + a_{12}x_2 \amp + \dots \amp + a_{1n}x_n \amp = b_1\\
a_{21}x_1 \amp + a_{22}x_2 \amp + \dots \amp + a_{2n}x_n \amp = b_2\\
\vdots \quad \amp \qquad \vdots \amp \amp \qquad \vdots \amp \vdots\\
a_{m1}x_1 \amp + a_{m2}x_2 \amp + \dots \amp + a_{mn}x_n \amp = b_m
\end{alignat*}
where \(a_{ij}, b_i \in \mathbb{R}\) are the coefficients of the system.
Here, the coefficients \(a_{ij}\) are written using double subscript notation. Here, \(a_{ij}\) represents the coefficients of \(x_j\) in the \(i\)th equation. The first subscript represents the equation number, and the second represents the variable number that it is a coefficient of.
- A system of linear equations is also sometimes called simultaneous linear equations, or simply a linear system, or sometimes simply a system.
- The variables \(x_1, \dots, x_n\) are sometimes called unknowns.
- A solution of a linear system is an ordered tuple \((s_1, \dots, s_n)\) which, when substituted for \((x_1, \dots, x_n)\text{,}\) respectively, makes each equation a true statement.
- The set of all possible solutions of a linear system is called its solution set.
