This section is going to be mostly devoted to giving the
induced matrices for a variety of standard linear transformations. We will be working exclusively with linear
transformations of the form 
and and for the most part we’ll be providing
equations and sketches of the transformations in but we’ll just be providing equations for the cases.
Let’s start this section out with two of the transformations
we looked at in the previous section just so we can say we’ve got all the main
examples here in one section.
Zero Transformation
In this case every vector x is mapped to the zero vector and so the transformation is,
and the induced matrix is the zero matrix, 0.
Identity
Transformation
The identity transformation will map every vector x to itself. The transformation is,
and so the induced matrix is the identity matrix.
Reflections
We saw a variety of reflections in in the previous section so we’ll give those
again here again along with some reflections in so we can say that we’ve got them all in one
place.
Note that in the when we say we’re reflecting about a given
plane, say the xy-plane, all we’re
doing is moving from above the plane to below the plane (or vise-versa of
course) and this means simply changing the sign of the other variable, z in the case of the xy-plane.
Orthogonal
Projections
We first saw orthogonal
projections in the section on the dot product. In that section we looked at projections only
in the ,
but as we’ll see eventually they can be done in any setting. Here we are going to look at some special
orthogonal projections.
Let’s start with the orthogonal projections in . There are two of them that we want to look
at. Here is a quick sketch of both of
these.
So, we project x
onto the x-axis or y-axis depending upon which we’re
after. Of course we also have a variety
of projections in as well.
We could project onto one of the three axes or we could project onto one
of the three coordinate planes.
Here are the orthogonal projections we’re going to look at
in this section, their equations and their induced matrix.
Contractions &
Dilations
These transformations are really just fancy names for scalar
multiplication, ,
where c is a nonnegative scalar. The transformation is called a contraction if and a dilation
if . The induced matrix is identical for both a
contraction and a dilation and so we’ll not give separate equations or induced
matrices for both.
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Contraction/Dilation
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Equations
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Induced Matrix
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Contraction/Dilation in
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Contraction/Dilation in
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Rotations
We’ll start this discussion in . We’re going to start with a vector x and we want to rotate that vector
through an angle in the counter-clockwise manner as shown
below.
Unlike the previous transformation where we could just write
down the equations we’ll need to do a little derivation work here. First, from our basic knowledge of
trigonometry we know that
and we also know that
Now, through a trig formula we can write the equations for w as follows,
Notice that the formulas for x and y both show up in
these formulas so substituting in for those gives,
Finally, since is a fixed angle and are fixed constants and so there are our
equations and the induced matrix is,
In we also have rotations but the derivations are
a little trickier. The three that we’ll
be giving here are counter-clockwise rotation about the three positive
coordinate axes.
Here is a table giving all the rotational equation and
induced matrices.
Okay, we’ve given quite a few general formulas here, but we
haven’t worked any examples with numbers in them so let’s do that.
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Example 1 Determine
the new point after applying the transformation to the given point. Use the induced matrix associated with each
transformation to find the new point.
(a) reflected about the xz-plane.
(b) projected on the x-axis.
(c) projected on the yz-plane.
Solution
So, it would be easier to just do all of these directly
rather than using the induced matrix, but at least this way we can verify
that the induced matrix gives the correct value.
(a) Here’s the
multiplication for this one.
So, the point maps to under this transformation.
(b) The
multiplication for this problem is,
The projection here is
(c) The
multiplication for the final transformation in this set is,
The projection here is .
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Let’s take a look at a couple of rotations.
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Example 2 Determine
the new point after applying the transformation to the given point. Use the induced matrix associated with each
transformation to find the new point.
(a) rotated in the counter-clockwise direction.
(b) rotated in the counter-clockwise direction about the
y-axis.
(c) rotated in the counter-clockwise direction about the
z-axis.
Solution
There isn’t much to these other than plugging into the
appropriate induced matrix and then doing the multiplication.
(a) Here is the
work for this rotation.
The new point after this rotation is then, .
(b) The matrix
multiplication for this rotation is,
The point after this rotation becomes . Note that we could have predicted this
one. The original point was in the yz-plane (because the x component is zero) and a counter-clockwise rotation about the y-axis would put the new point in the xy-plane with the z component becoming the x
component and that is exactly what we got.
(c) Here’s the
work for this part and notice that the angle is not one of the “standard”
trig angles and so the answers will be in decimals.
The new point under this rotation is then .
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Finally, let’s take a look at some compositions of
transformations.
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Example 3 Determine
the new point after applying the transformation to the given point. Use the induced matrix associated with each
transformation to find the new point.
(a) Dilate
by 2 (i.e.
2x) and then project on the y-axis.
(b) Project
on the y-axis
and then dilate by 2.
(c) Project
on the x-axis
and then rotate by counter-clockwise.
(d) Rotate
counter-clockwise and then project on the x-axis.
Solution
Notice that the first two are the same translations just
done in the opposite order and the same is true for the last two. Do you expect to get the same result from
each composition regardless of the order the transformations are done?
Recall as
well that in compositions we can get the induced matrix by multiplying the
induced matrices from each transformation from the right to left in the order
they are applied. For instance the
induced matrix for the composition is BA
where is the first transformation applied to the
point.
(a) Dilate by
2 (i.e. 2x) and then project on the
y-axis.
The induced matrix for this composition is,
The matrix multiplication for the new point is then,
The new point is then .
(b) Project on
the y-axis and then dilate by 2.
In this case the induced matrix is,
So, in this case the induced matrix for the composition is
the same as the previous part.
Therefore, the new point is also the same, .
(c) Project on
the x-axis and the rotate by counter-clockwise.
Here is the induced matrix for this composition.
The matrix multiplication for new point after applying
this composition is,
The new point is then,
(d) Rotate counter-clockwise and then project on the x-axis.
The induced matrix for the final composition is,
Note that this is different from the induced matrix from (c) and so we should expect the new
point to also be different. The fact
that the induced matrix is different shouldn’t be too surprising given that
matrix multiplication is not a commutative operation.
The matrix multiplication for the new point is,
The new point is then, and as we expected it was not the same as
that from part (c).
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So, as this example has shown us transformation composition is
not necessarily commutative and so we shouldn’t expect that to happen in most
cases.