Overview on linear systems

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Our journey through linear algebra begins with linear systems.

A linear function in one variable is of the form $f(x) = ax+b$ where $x$ is a variable and $a,b$ are numbers (or scalars as they are referred to in linear algebra). It is homogeneous if $b=0$ . Similarly, a linear function in n variables is one of the form $f(x_1,x_2,\dots ,x_n) = a_1x_1 + a_2x_2 + \dots a_nx_n + b$ where the $x_i$ are variables (or unknowns) and the $a_i$ are scalars. Again, the linear function is called homogeneous if the constant term $b$ is zero. Next, a linear equation in n variables is one of the form $a_1x_1 + a_2x_2 + \dots a_nx_n = b$ where the expression on the left is a linear homogeneous function in $n$ variables, and the term on the right is a constant.

Consider $\begin{equation} 3x_1 - 7x_2 + 11x_3 = 14. \end{equation}$ This is a linear equation in three variables: $x_1, x_2$ , and $x_3$ . A solution to a linear equation is an assignment of values to each of the variables appearing which makes the equation hold true.

We see $x_1 = 1, x_2 = 0, x_3 = 1$ is a solution to this linear equation, because when substituted into the left-hand side it results in the value 14.

A system of equations is a collection of linear equations in the same set of variables, or unknowns $\begin{alignat*} {5}\tag{2}\label{eqn:sys} a_{11}x_1 &&+ a_{12}x_2 && + {}\ldots{} && + a_{1n}x_n && = b_1 &\\ a_{21}x_1 + && a_{22}x_2 + && {}\ldots{} && + a_{2n}x_n && = b_2 &\\ \vdots && \vdots && {}\ldots{} && \vdots && \vdots &\\ a_{m1}x_1 &&+ a_{m2}x_2 && + {}\ldots{} && + a_{mn}x_n && = b_m & \end{alignat*}$

and a solution to that system is an assignment of values to the variables which make each equation hold true. The solution set of a system of equations is the collection, or set of all the solutions of that system (which will be empty if the system has no solutions). Given $m$ equations in the same set of $n$ unknowns, the ordered pair of integers $(m,n)$ are referred to as the dimensions of the system, and the system would be referred to as an $m\times n$ system of equations (read: m-by-n system of equations), or an $m\times n$ system for short.

Systems of equations, or even a single equation, need not have a solution. To illustrate, consider the equation $0x_1 + 0x_2 + 0x_3 = 4$ Obviously, any set of values substituted into the left-hand side of this equation will produce the value zero, which is not equal to 4. But even non-zero systems might not have a solution. As an example, consider the $\begin{alignat*} {3} x_1 && - 2x_2 && =& \phantom{1}7 \\ 2x_1 && - 4x_2 && =& 16 \end{alignat*}$

The left-hand side of the second equation is twice that of the first. So if we take any solution of the first equation and plug in those values on the left-hand side of the second equation, we will always get $2*7 = 14\ne 16$ . Therefore, these two equations will never be simultaneously satisfied, and so the system doesn’t have a solution.

A system which has at least one solution is called consistent. If it doesn’t have any solutions it is called inconsistent.

In the following activities, the main questions we need to answer are:

Given a system of equations, does it have one or more solutions (in other words is it consistent)?
If it is consistent, what are the solutions?
Is there a systematic way of finding the solutions?

The material in the first module is focused on answering these questions.