Search for notes by fellow students, in your own course and all over the country.
Browse our notes for titles which look like what you need, you can preview any of the notes via a sample of the contents. After you're happy these are the notes you're after simply pop them into your shopping cart.
Document Preview
Extracts from the notes are below, to see the PDF you'll receive please use the links above
Geometrical Transformations
3
...
They
also have many important applications
...
For example, structural engineers use stretches,
shears, and rotations to understand the deformation of materials
...
Many of
these simple geometrical transformations in R2 and R3 are linear
...
(projil and perpil belong to the list of geometrical transformations, too, but they were
discussed in Chapter 1 and so are not included here
...
See Figure 3
...
1
...
Thus, Re(t1) = tRe(1) for any t E R
...
Since the shape
of a parallelogram is not altered by a rotation, the picture for the parallelogram rule of
addition should be unchanged under a rotation, so Re(1 + y) = Re(1) + Re(Y)
...
3
...
Counterclockwise rotation through angle e in the plane
...
Assuming that
calculate
[Re]
...
3
...
3
...
What is the matrix of rotation of�
...
/3/2
-
v'3/2
-1/2
]
X1
EXERCISE 1
Determine
[Rrr14]
and use it to calculate
Rrr14(1, 1)
...
Rotation Through Angle(} About the x3-axis in IR
...
3
...
This rotation leaves x3 unchanged, so that if the transformation is de
R, R(O,
noted
then
0, 1)
(0, 0, 1)
...
Is it
3
possible to determine the matrix of a rotation about an arbitrary axis in JR ? We shall
see how to do this in Chapter 7
...
3
...
Stretches Imagine that all lengths in the x1 -direction in the plane are stretched by a
scalar factor t > 0, while lengths in the x2-direction are left unchanged (Figure 3
...
4)
...
) It should be obvious that
stretches can also be defined in the x2-direction and in higher dimensions
...
Contractions and Dilations If a linear operator T : JR2
[� �l
with t > 0, then for any 1, T(x)
=
�
JR2 has matrix
tx, so that this transformation stretches
vectors in all directions by the same factor
...
If 0 < t < 1, such a
transformation is called a contraction; if t > l, it is a