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Systems of Linear EquationsTypically when we encounter a system of simultaneous linear equations, we have n equations in n unknown variables. Generally the most useful of these have a unique solution. As an example, the following set of equations:
has a solution: x = 3, y = −1, and z = 2. In this situation with 3 equations in 3 variables, each equation represents a plane in a three-dimensional space. The solution is a point which is the intersection of the three planes. We know that there are other situations. In general, a set of n simultaneous linear equations in n variables may have:
Let us explore these situations, along with the the more normal case with same number of unknowns and equations. Fewer equations than variablesThere is no one-dimensional case with one variable since this would imply there are zero equations!
For the two-dimensional case (with 2 variables), we only have the situation of one equation if the number of equations is less than the number of variables.
As an example:
The solution set contains all possible points that satisfies this equation. In this case the solution set is all the points on the line: y = x − 2 We have infinitely many solutions. Another way to represent this is:
For a three-dimensional situation, we have three variables, and if the number of equations is less than three, we get two cases: one equation or two equations! An example of a single equation with three variables is:
This represents a plane in three-dimensional space with intercepts (points where it crosses the three axes): (3,0,0), (0,9/4,0), and (0,0,9/2). Clearly this is a situation where there are infinitely many solutions! In general, 1 equation in n variables with n > 1 will also give infinitely many solutions. Adding another equation to the first one, to give us a system of 2 equations in the 3 variables, can give us several possibilities. Consider:
The second plane is not coincident with the first but is parallel to it. Therefore, we get no solutions. Adding more equations to the set cannot change this. A slightly more interesting case is:
The second plane is coincident with the first. Therefore, we get an infinitely many solutions. A most interesting case is an equation that represents a plane that intersects the first in a straight line:
Because the solution set is all the points on the straight line, we get infinitely many solutions. However, this situation holds the promise that an additional equation representing a plane that intersects the line at a point might yield a unique solution for the complete 3 equation set in the 3 variables! We can generalize these:
Same number of equations as variablesThis is the case we normally study. We know that there are three possible outcomes:
x = 5
This has a unique solution! It is only if we permit the generalization of 0•x = 3 that we could get no solutions. And only if we consider something like 0•x = 0 that we could get an infinite number of solutions. Of course, one could argue that for these two rather strange situations there is no variable rather than one and the problem is not really one-dimensional! For practical problems, we would never encounter these two situations, but mathematicians tend to exam all possible cases when generalizing. In any case, we will simply consider that if we have one equation in one unknown, then there is a unique solution.
In two-dimensions with two variables, each equation represents a line which we can graph in on ordinary xy-plane. Therefore, we find:
For three-dimensions with three variables, each equation represents a plane. Therefore, we find:
While these are the geometric picture, fortunately we do not generally need to discern the exact situation that leads to either 2. no solution or 3. infinitely many solutions. We discover that in solving the system we get an inconsistency (no solution) or we arrive at an identity where the coefficients of all the variables are zero and the constant is zero (infinitely many solutions)! With four-dimensions and higher it gets harder to visualize since each equation represents a three-dimensional space embedded in a four-dimensional hyperspace, but the geometric principles are the same. And, we generally are not interested in why a particular situation falls into one of the three categories. Fortunately, the algebraic and arithmetic manipulations leads us directly there.
More equations than variablesIf too few equations gives us an under determined problem, then we might suspect that too many might give an over determined situation such that there might not a unique solution. This is indeed the case!Let us start with the simplest case: two equations with only one variable (this is the one-dimensional case):
x = 5
2x = 10 Clearly this system has a unique solution, x = 5, but it is also a very special case. The two equations are not independent. On the other hand, more generally we might find something like:
x = 5
2x = 12 This system has no solution. However, since the second equation has the solution x = 6 we might consider a value such as x = 5.5 (the average between the two individual solutions) as in some way representing the "best" answer if you insist upon a specific result. Adding more equations cannot change this general picture. It is only with great luck or judicious choice that a set of two or more equations in one variable will yield a unique solution. Generally, we simply get no solution, or perhaps a "best" solution by averaging the several different ones together. The next more complex situation has three equations with one or two variables. Let us start with a two-dimensional case (with two unknown variables) that has two equations (then we will explore what happens when we add another equation):
These are two intersecting lines with a solution x = 3 and y = 1. If we add a third equation such as:
we discover that it also passes through the point (3,1). That is, these three particular lines all pass through the same point. This is a very special case. What luck! In general, the new third line will probably not pass through the intersection of the first two. Indeed, the three lines will probably come together and define a triangle with three vertices formed by the intersection of the lines for equations 1 and 2, equations 1 and 3, and equations 2 and 3. Of course, the third line might be parallel to either of the first two or even coincident with one of them. We already know this leads to problems! What can a person do? There is a procedure that can provide a best solution under some of these circumstances. Clearly if the triangle formed by the three lines is small, then a point in its middle must represent a "good solution" even though it is not an exact solution. The "best" can be specified in a least-squares sense that is similar to linear regression. On the other hand, if we have a case in two-dimensions with three parallel non-coincident lines; that is, not even one pairwise intersection, then there is not even a best solution to this no solution case. Similar arguments apply to higher dimensional cases. In three dimensions, each equation represents a plane. If we have four such equations, then two (or three or four) might be parallel and we get no solution. The four planes might come together in a unique point—if we are really lucky! More generally, the four planes may enclose a small volume. A point near the middle of this volume could represent a best solution.
SummaryTo summarize possible outcomes for the first few possible cases where the red-shaded entries represent over-determined cases and the blue-shaded entries are under-determined:
The same pattern continues for even larger systems of simultaneous linear equations. We can draw some generalizations:
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© 1999, 2000, 2001, 2002, 2003 by Lawrence Turner |
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