Question

Let ? denotes the counterclockwise rotation through 60 degrees, followed by reflection in the line ?=?.

(i) Show that ? is a linear transformation.

(ii) Write it as a composition of two linear transformations.

(iii) Find the standard matrix of ?.

Answer #1

Find the matrix of the reflection of R2 across the line y =x/3
followed by the reflection of R2 across the line y = x/2 What type
of transformation of the plane is this composition?
thank you.

. In this question we will investigate a linear transformation F
: R 2 → R 2 which is defined by reflection in the line y = 2x. We
will find a standard matrix for this transformation by utilising
compositions of simpler linear transformations. Let Hx be the
linear transformation which reflects in the x axis, let Hy be
reflection in the y axis and let Rθ be (anticlockwise) rotation
through an angle of θ. (a) Explain why F =...

* Consider the transformations T1=‘reflection across the x-axis’
and T2=‘reflection across the line y = x’. (a) Find the matrices A1
and A2 corresponding to T1 and T2, respectively. (b) Show that (A1)
2 = I, and give a geometrical interpretation of this. (c) Use
matrix multiplication to find the geometric effect of T1 followed
by T2, showing all your reasoning. (d) The product T (θ)T (φ) of
any two reflections T (θ) and T (φ) with angles θ and...

(a) Find the matrix of the reflection of R^2 across the line y =
(1 / 3)x followed by the reflection of R^2 across the line y =
(1/2) x. What type of transformation of the plane is this
composition?
b) Find the principal axes y1 and y2 diagonalizing the quadratic
form q = (x^2)1 + (8)x1x2 + (x^2)2

1.1. Let R be the counterclockwise rotation by 90 degrees.
Vectors r1=[3,3] and r2=[−2,3] are not perpendicular. The inverse U
of the matrix M=[r1,r2] has columns perpendicular to r2 and r1, so
it must be of the form U=[x⋅R(r2),y⋅R(r1)]^T for some scalars x and
y. Find y^−1.
1.2. Vectors r1=[1,1] and r2=[−5,5] are perpendicular. The
inverse U of the matrix M=[r1,r2] has columns perpendicular to r2
and r1, so it must be of the form U=[x⋅r1,y⋅r2]^T for some scalars
x...

Let T be the linear transformation from R2 to R2, that rotates a
vector clockwise by 60◦ about the origin, then reﬂects it about the
line y = x, and then reﬂects it about the x-axis.
a) Find the standard matrix of the linear transformation T.
b) Determine if the transformation T is invertible. Give detailed
explanation. If T is invertible, ﬁnd the standard matrix of the
inverse transformation T−1.
Please show all steps clearly so I can follow your...

Ahmadi, Inc. manufactures laptop and desktop computers. In the
upcoming production period, Ahmadi needs to decide how many of each
type of computers should be produced to maximize profit. Each
computer goes through two production processes. Process
I, involves assembling the circuit boards and
process II is the installation of the
circuit boards into the casing. Each laptop requires 24 minutes of
process I time and 16 minutes of process II time. Each desktop
requires 8 minutes of process I...

Questions 1 through
6 work with the length of the sidereal year vs. distance from the
sun. The table of data is shown below.
Planet
Distance from Sun
(in millions of
miles)
Years (as a
fraction of Earth
years)
ln(Dist)
ln(Year)
Mercury
36.19
0.2410
3.5889
-1.4229
Venus
67.63
0.6156
4.2140
-0.4851
Earth
93.50
1.0007
4.5380
0.0007
Mars
142.46
1.8821
4.9591
0.6324
Jupiter
486.46
11.8704
6.1871
2.4741
Saturn
893.38
29.4580
6.7950
3.3830
Uranus
1,794.37
84.0100
7.4924
4.4309
Neptune
2,815.19
164.7800
7.9428...

1. An unstandardized test of aggression has a mean of 35.19. You
take a sample of 44 video game players and find their scores on
this test have a mean of 39.84 and a standard deviation of 12.42.
Are their aggression scores greater than the standard of 35.19?
a. State the null and alternative hypotheses.
b. Find your critical value(s) with a criterion of .01. Remember
if the appropriate line is not available exactly in the table to
pick the...

The British Department of Transportation studied to see if
people avoid driving on Friday the 13th.
They did a traffic count on a Friday and then again on a Friday the
13th at the same two locations ("Friday the 13th,"
2013). The data for each location on the two different dates is in
following table:
Table: Traffic Count
Dates
6th
13th
1990, July
139246
138548
1990, July
134012
132908
1991, September
137055
136018
1991, September
133732
131843
1991, December
123552...

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