# Solution Sets for Systems of Linear Equations

For a system of equations with r equations and k unknowns,
one can have a

number of different outcomes. For the sake of visualization, consider the case

of r equations in three variables. Geometrically, then, each of our equations

is the equation of a plane in three-dimensional space. To find solutions to

the system of equations, we look for the common intersection of the planes

(if an intersection exists). Here we have five different possibilities:

1. **No solutions.** Some of the equations are contradictory, so no solu-

tions exist.

2. **Unique Solution.** The planes have a unique point of intersection.

3. **Line.** The planes intersect in a common line; any point on that line

then gives a solution to the system of equations.

4.** Plane.** Perhaps you only had one equation to begin with, or else all

of the equations coincide geometrically. In this case, you have a plane

of solutions, with two free parameters.

5. **All of** R^{3}. If you start with no information, then any point in R^{3} is a

solution. There are three free parameters.

In general, for systems of equations with k unknowns , there are k + 2

possibile outcomes, corresponding to the number of free parameters in the

solutions set, plus the possibility of no solutions. These types of solution

sets are hard to visualize, but luckily 'hyperplanes' behave like planes in R^{3}

in many ways.

**Non-Leading Variables**

Variables that are not a pivot in the reduced row echelon form of a linear

system are free. Se set them equal to arbitrary parameters
.

**Example**

Here, and are the pivot variables and
and are non-leading

variables, and thus free. The solutions are then of the form

The preferred way to write a solution set is with set
notation. Let S be

the set of solutions to the system. Then:

It's worth noting that if we knew how to multiply matrices
of any size,

we could write the previous system as MX = v, where

Given two vectors we can add them term -by-term:

We can also multiply a vector by a scalar, like so:

Then yet another way to write the solution set for the example is:

where

**Definition** Let X and Y by vectors and α and β be scalars. A
function f

is linear if

Eventually, we'll prove that matrix multiplication is linear. Then we will

know that:

Then the two equations MX = v and together

say that:

for any .

Choosing , we obtain
. Given the particular

solution to the system, we can then deduce that

is an example of a particular solution to the system.

Setting, and recalling the particular solution
,

we obtain .

Likewise, setting , we obtain .

and
are homogeneous solutions to the system.

**Example** Consider the linear system with the augmented matrix we've

been working with.

Recall that the system has the following solution set:

Then
says that solves
the system , which is

certainly true.

says thatsolves the system.

says that solves
the system.

**Definition** Let M a matrix and V a vector. Given the linear system MX =

V , we call
a particular solution if
. We call Y a homogeneous

solution if MY = 0.

The linear system MX = 0 is called the (associated) homogeneous sys-

tem.

If is a particular solution, then the general solution to the system is:

In other words, the general solution = particular + homogeneous.

Review Questions

1. Write down examples of augmented matrices corresponding to each

of the five types of solution sets for systems of equations with three

unknowns.

2. Let

Propose a rule for MX so that MX = 0 is equivalent to the
linear

system:

Does your rule for multiplying a matrix times a vecotr
obey the lin-

earity property ? Prove it!

3. The standard basis vector e_{i} is a column vector with a
one in the

ith row, and zeroes everywhere else. Using the rule for multiplying

a matrix times a vector in the last problem, find a simple rule for

multiplying Me_{i}, where M is the general matrix defined in the last

problem.

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