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Linear Algebra

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Euclidean n-Space E-Book   Chapter   Section Examples of Linear Transformations

In this section we’re going to take a look at a special kind of function that arises very naturally in the study of Linear Algebra and has many applications in fields outside of mathematics such as physics and engineering.  This section is devoted mostly to the basic definitions and facts associated with this special kind of function.  We will be looking at a couple of examples, but we’ll reserve most of the examples for the next section.

 

Now, the first thing that we need to do is take a step back and make sure that we’re all familiar with some of the basics of functions in general.  A function, f, is a rule (usually defined by an equation) that takes each element of the set A (called the domain) and associates it with exactly one element of a set B (called the codomain).  The notation that we’ll be using to denote our function is

 

 

 

 

When we see this notation we know that we’re going to be dealing with a function that takes elements from the set A and associates them with elements from the set B.  Note as well that it is completely possible that not every element of the set B will be associated with an element from A.  The subset of all elements from B that are associated with elements from A is called the range.

 

In this section we’re going to be looking at functions of the form,

 

 

In other words, we’re going to be looking at functions that take elements/points/vectors from  and associate them with elements/points/vectors from .  These kinds of functions are called transformations and we say that f maps  into .  On an element basis we will also say that f maps the element u from  to the element v from .

 

So, just what do transformations look like?  Consider the following scenario.  Suppose that we have m functions of the following form,

 

 

 

Each of these functions takes a point in , namely , and maps it to the number .  We can now define a transformation  as follows,

 

 

In this way we associate with each point  from  a point  from  and we have a transformation.

 

Let’s take a look at a couple of transformations.

 

 

Example 1  Given

            

define  as,

                                   

 

Note that the second form is more convenient since we don’t actually have to define any of the w’s in that way and is how we will define most of our transformations.

 

We evaluate this just as we evaluate the functions that we’re used to working with.  Namely, pick a point from  and plug into the transformation and we’ll get a point out of the function that is in .  For example,

                                                   

 

Example 2  Define  as .  A sample evaluation of this transformation is,

                                                        

 

Now, in this section we’re going to be looking at a special kind of transformation called a linear transformation.  Here is the definition of a linear transformation.

 

Definition 1  A function  is called a linear transformation if for all u and v in  and all scalars c we have,

                               

 

We looked at two transformations above and only one of them is linear.  Let’s take a look at each one and see what we’ve got.

 

Example 3  Determine if the transformation from Example 2 is linear or not.

 

Solution

Okay, if this is going to be linear then it must satisfy both of the conditions from the definition.  In other words, both of the following will need to be true.

                                          

 

                                  

 

In this case let’s take a look at the second condition.

                                           

 

The second condition is not satisfied and so this is not a linear transformation.  You might want to verify that in this case the first is also not satisfied.  It’s not too bad, but the work does get a little messy.

 

Example 4  Determine if the transformation in Example 1 is linear or not.

 

Solution

To do this one we’re going to need to rewrite things just a little.  The transformation is defined as  where,

                                                              

 

Now, each of the components are given by a system of linear (hhmm, makes one instantly wonder if the transformation is also linear…) equations and we saw in the first chapter that we can always write a system of linear equations in matrix form.  Let’s do that for this system.

 

                                     

 

Now, notice that if we plug in any column matrix x and do the matrix multiplication we’ll get a new column matrix out, w.  Let’s pick a column matrix x totally at random and see what we get.

                                                      

 

Of course, we didn’t pick x completely at random.  Notice that x we choose was the column matrix representation of the point from  that we used in Example 1 to show a sample evaluation of the transformation.  Just as importantly notice that the result, w, is the matrix representation of the point from  that we got out of the evaluation.

 

In fact, this will always be the case for this transformation.  So, in some way the evaluation  is the same as the matrix multiplication  and so we can write the transformation as

                                                                 

Notice that we’re kind of mixing and matching notation here.  On the left x represents a point in   and on the right it is a  matrix.  However, this really isn’t a problem since they both can be used to represent a point in .  We will have to get used to this notation however as we’ll be using it quite regularly.

 

Okay, just what were we after here.  We wanted to determine if this transformation is linear or not.  With this new way of writing the transformation this is actually really simple.  We’ll just make use of some very nice facts that we know about matrix multiplication.  Here is the work for this problem

                                    

                                                

 

So, both conditions of the definition are met and so this transformation is a linear transformation.

 

There are a couple of things to note here.  First, we couldn’t write the transformation from Example 2 as a matrix multiplication because at least one of the equations (okay both in this case) for the components in the result were not linear.

 

Second, when all the equations that give the components of the result are linear then the transformation will be linear.  If at least one of the equations are not linear then the transformation will not be linear either.

 

Now, we need to investigate the idea that we used in the previous example in more detail.  There are two issues that we want to take a look at. 

 

First, we saw that, at least in some cases, matrix multiplication can be thought of as a linear transformation.  As the following theorem shows, this is in fact always the case.