In this section we'll look much more closely at the “systems of equations” approach to linear algebra.

First a few words about notation. When there are seventeen variables in a problem it becomes really awkward to use different letters for each variable. When there are a thousand variables it's impossible! We will follow the almost universal convention that the letter \(x\) will be used for the variables, with a subscript to identify which one. If we were to translate the problem from Section 1.1 into this notation it would become \begin{gather*} x_1 + x_2 = 42 \\ x_1 - x_2 = 6. \end{gather*}

A linear combination of some set of numbers \(\{ x_1, x_2, \ldots, x_n \}\) is created by multiplying each of the \(x\)'s by constants and then adding everything up. Of course if the constants are \(1\) or \(-1\) (as in the previous example) we tend to forget that they're there!

An equation is linear if it has the form of a linear combination set equal to some value on the right-hand side -- or if it can be put into that form. For example \begin{gather*} x_1 + 2x_2 + 3x_3 + 4x_4 + 5x_5 = 15 \end{gather*} is a linear equation in five variables.

Also, \begin{gather*} x_1 + 3x_3 = x_2 + x_4 \end{gather*} is a linear equation (in four variables) because we can manipulate it into the form \begin{gather*} x_1 - x_2 + 3x_3 - x_4 = 0. \end{gather*}

A system of equations is just a collection of linear equations.

The notation for systems of equations gets a bit complicated when we try to write them in general (that is, without particular values given for the various constants involved). There are three sorts of things that need names in such a system: the variables, the coefficients of the variables, and the numbers on the right-hand sides. There is a convention that is fairly universal for the names and numbering of these elements. The variables are \(x\)'s with subscripts, the right-hand sides are \(b\)'s with subscripts, and the coefficients are \(a\)'s with two subscripts (we need to indicate the equation that a given coefficient is in and also, which variable it is multiplying.

For example, here is how we would write the general form of a system of three equations in four unknowns: \begin{equation*} \begin{alignedat}{5} \aij{1}{1} x_1 \amp {}+{} \amp \aij{1}{2} x_2 \amp {}+{} \amp \aij{1}{3} x_3 \amp {}+{} \amp \aij{1}{4} x_4 \amp {}={} \amp b_1 \\ \aij{2}{1} x_1 \amp {}+{} \amp \aij{2}{2} x_2 \amp {}+{} \amp \aij{2}{3} x_3 \amp {}+{} \amp \aij{2}{4} x_4 \amp {}={} \amp b_2 \\ \aij{3}{1} x_1 \amp {}+{} \amp \aij{3}{2} x_2 \amp {}+{} \amp \aij{3}{3} x_3 \amp {}+{} \amp \aij{3}{4} x_4 \amp {}={} \amp b_3 \\ \end{alignedat} \end{equation*} Notice that the indices on the \(x\)'s run from 1 to 4, the indices on the \(b\)'s run from 1 to 3, and that there are a total of 12 coefficients.

A bit more formalism is appropriate now. We'll start with some definitions.

What does it mean to say we have found an “answer” to a system of equations? Essentially, it is this: we have found a set of values for the variables that “work” in all of the equations. Sometimes people say that this set of values “satisfies” the equations. To be completely clear, what is meant is that if one substitutes these values for the variables in the equations of the system, all of them (the equations) will be true. It is convenient to regard such a set of values as a vector. For example the solution we obtained in Example 1.2.3 would be regarded as the vector \(\langle 225, 167, 333 \rangle\).

The general idea is this: there are lots and lots of different linear systems that are equivalent. They all have the same solution set. Some of these systems are in a nice form that allows us to see what the solution set is. Others are not. We need to transform the latter into the former!

More specifically, there are three operations that can be applied to linear systems which do not have any effect on solution sets. We can apply these three operations in any way we like! We'll just be transforming our linear system into a slightly different one that is equivalent to the original. Finally, you'll see that it is pretty easy to strategize a bit and transform difficult linear systems into the nice sort (where the solution set is very evident) using these three operations.

The three operations go by many names; we'll refer to them as Reordering, Scaling and Combining. In the next few paragraphs we'll discuss each of them in turn and explain why they don't have an effect on the solution set of a system.

Reordering means what it sounds like. The solution set is determined by checking whether a given solution vector satisfies all of the equations. It is pretty clear that the order that the equations are listed in is of little importance. In many treatments of linear algebra an operation called “swapping” is used instead — swapping two equations is a special (particularly simple) instance of reordering and any more general reordering can be accomplished by a succession of swaps.

Scaling is another operation where it is fairly obvious that there will be no effect on solution sets. Scaling involves multiplying both sides of an equation by some non-zero constant. Very often that non-zero constant will be the reciprocal of the coefficient of one of the variables; scaling by such a constant is useful in solving for that variable. Perhaps it is clear that multiplying both sides of an equation by the same thing will have no impact on what values of the variables satisfy the equation… But why does the constant need to be non-zero? Multiplying both sides of any equation by \(0\) will produce a new equation that looks like \(0 = 0\) which is certainly true! In fact, of course, that's what the problem is; if the equation was previously false for some vector of variable values (thus it served to exclude that vector from the solution set) it will now be true. So vectors of variable values that previously were not in the solution set will now be in it — that's the sort of thing we are trying to avoid!

Combining (a.k.a replacement) is the most difficult of the three operations and as you might guess, it is also the most useful. Combining consists of adding a multiple of some other equation to a given one. Another way to think of this is that we replace some equation by itself plus a multiple of some other equation. This is probably why some people call this operation Replacement.

When we added the equations \(x+y=42\) and \(x-y=6\), obtaining the new equation \(2x=48\) back in Section 1.1 we were really doing a “Combining” operation. By the way, when we divided both sides of that new equation by \(2\) we were “Scaling.”

We'll close this section by giving an example — using the three operations to find the solution of a small linear system.