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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.
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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,
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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.
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.
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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.
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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.
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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.
Proof : The proof
here is really simple and in fact we pretty much saw it last example.
So, the induced function, ,
satisfies both the conditions in the definition of a linear transformation and
so it is a linear transformation.
So, any time we do matrix multiplication we can also think
of the operation as evaluating a linear transformation.
The other thing that we saw in Example 4 is that we were
able, in that case, to write a linear transformation as a matrix
multiplication. Again, it turns out that
every linear transformation can be written as a matrix multiplication.
Proof : First
let,
be the standard
basis vectors for and define A
to be the matrix whose ith column is . In other words, A is given by,
Next let x be any
vector from . We know that we can write x in terms of the standard basis
vectors as follows,
In order to prove this theorem we’re going to need to show
that for any x (which we’ve got a
nice general one above) we will have . So, let’s start off and plug x into T using the general form as written out above.
Now, we know that T
is a linear transformation and so we can break this up at each of the “+”’s as
follows,
Next, each of the ’s are scalars and again because T is a linear transformation we can
write this as,
Next, let’s notice that this is nothing more than the
following matrix multiplication.
But the first matrix nothing more than A and the second is just x
and we when we define A as we did
above we will get,
and so we’ve proven what we needed to.
In this proof we used the standard basis vectors to define
the matrix A. As we will see in a later chapter there are
other choices of vectors that we could use here and these will produce a
different induced matrix, A, and we
do need to remember that. However, when
we use the standard basis vectors to define A,
as we’re going to in this chapter, then we don’t actually need to evaluate T at each of the basis vectors as we did
in the proof. All we need to do is what
we did in Example 4, write down the coefficient matrix for the system of
equations that we get by writing out each of the components as individual
equations.
Okay, we’ve done a lot of work in this section and we
haven’t really done any examples so we should probably do a couple of
them. Note that we are saving most of
the examples for the next section, so don’t expect a lot here. We’re just going to do a couple so we can say
we’ve done a couple.
So, the two examples above are standard examples and we did
need them taken care of. However, they
aren’t really very illustrative for seeing how to construct the matrix induced
by the transformation. To see how this
is done, let’s take a look at some reflections in . We’ll look at reflections in in the next section.
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Example 7 Determine
the matrix induced by the following reflections.
(a) Reflection
about the x-axis. [Solution]
(b) Reflection
about the y-axis. [Solution]
(c) Reflection
about the line . [Solution]
Solution
Note that all of these will be linear transformations of
the form .
(a) Reflection about the x-axis.
Let’s start off with a sketch of what we’re looking for
here.
So, from this sketch we can see that the components of the
for the translation (i.e. the
equations that will map x into w) are,
Remember that will be the first component of the
transformed point and will be the second component of the
transformed point.
Now, just as we did in Example 4 we can write down the
matrix form of this system.
So, it looks like the matrix induced by this reflection
is,
[Return to Problems]
(b) Reflection about the y-axis.
We’ll do this one a little quicker. Here’s a sketch and the equations for this
reflection.
The matrix induced by this reflection is,
[Return to Problems]
(c) Reflection about the
line .
Here’s the sketch and equations for this reflection.
The matrix induced by this reflection is,
[Return to Problems]
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Hopefully, from these examples you’re starting to get a feel
for how we arrive at the induced matrix for a linear transformation. We’ll be seeing more of these in the next
section, but for now we need to move on to some more ideas about linear
transformations.
Let’s suppose that we have two
linear transformations induced by the matrices A and B, and . If we take any x out of will map x
into . In other words, will be in and notice that we can then apply to this and its image will be in . In summary, if we take x out of and first apply to x
and then apply to the result we will have a transformation
from to .
This process is called composition
of transformations and is denoted as
Note that the order here is important. The first transformation to be applied is on
the right and the second is on the left.
Now, because both of our original transformations were
linear we can do the following,
and so the composition is the same as multiplication by BA.
This means that the composition will be a linear transformation provided
the two original transformations were also linear.
Note as well that we can do composition with as many
transformations as we want provided all the spaces correctly match up. For instance with three transformations we
require the following three transformations,
and in this case the
composition would be,
Let’s take a look at a couple of examples.
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Example 8 Determine
the matrix inducted by the composition of reflection about the y-axis followed by reflection about
the x-axis.
Solution
First, notice that reflection about the y-axis should change the sign on the x coordinate and following this by a
reflection about the x-axis should
change the sign on the y
coordinate.
The two transformations here are,
The matrix induced by the composition is then,
Let’s take a quick look at what this does to a point. Given x
in we have,
This is what we expected to get. This is often called reflection about the
origin.
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Example 9 Determine
the matrix inducted by the composition of reflection about the y-axis followed by another reflection
about the y-axis.
Solution
In this case if we reflect about the y-axis twice we should end right back where we started.
The two transformations in this case are,
The induced matrix is,
So, the composition of these two transformations yields
the identity transformation. So,
and the composition will not change the original x as we guessed.
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