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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 very 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 visa-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.