Question

. 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 = Rθ ◦ Hx ◦ R−θ for an appropriate choice of θ, and use this to calculate the standard matrix of F.

Answer #1

Can
someone explain linear transformations which rotates vectors by
certain degrees?
Examples:
R^3--> R^3: A linear transformation which rotates vectors
90 degrees about the x axis/y axis/z-axis (how would the matrix
look if about a different axis)
what if it rotates 180 degrees?
R^2-->R^2?

let let T : R^3 --> R^2 be a linear transformation defined by
T ( x, y , z) = ( x-2y -z , 2x + 4y - 2z) a give an example of two
elements in K ev( T ) and show that these sum i also an element of
K er( T)

Find the standard matrix for the linear transformation f(a, b,
c, d)=(b-c+d, 2b-3d).
Find the standard matrix for the linear transformation that
flips the xy plane over the y axis and rotates it by π/4 radians
CCW.

Find the matrix of the linear transformation which reflects
every vector across the y-axis and then rotates every vector
through the angle π/3.

(a) Let T be any linear transformation from R2 to
R2 and v be any vector in R2 such that T(2v)
= T(3v) = 0. Determine whether the following is true or false, and
explain why: (i) v = 0, (ii) T(v) = 0.
(b) Find the matrix associated to the geometric transformation
on R2 that first reflects over the y-axis and then
contracts in the y-direction by a factor of 1/3 and expands in the
x direction by a...

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...

We consider the plane region R delimited by the curves y = cos (x) and y = (x − π) ^ 2 −2.
(a) Determine the volume of the solid generated by the rotation of R revolves around the
right y = −3.
(b) Determine the volume of the solid generated by the rotation of R revolves around the
right x = 0.
For (a) and (b), observe the following procedure:
- Draw a sketch (2D) of the R region...

In this question we denote by P2(R) the set of functions {ax2 +
bx + c : a, b, c ∈ R}, which is a vector space under the usual
addition and scalar multiplication of functions. Let p1, p2, p3 ∈
P2(R) be given by p1(x) = 1, p2(x) = x + 2x 2 , and p3(x) = αx + 4x
2 . a) Find the condition on α ∈ R that ensures that {p1,
p2, p3} is a basis...

2.) Suppose we use a person's dad's height to predict how short
or tall the person will be by building a regression model to
investigate if a relationship exists between the two variables.
Suppose the regression results are as follows:
Least Squares Linear Regression of Height
Predictor
Variables Coefficient Std Error T P
Constant 20.2833 8.70520 2.33 0.0223
DadsHt 0.67499 0.12495 5.40 0.0002
R² 0.2673 Mean Square Error (MSE) 23.9235
Adjusted R² 0.2581 Standard Deviation
4.9000...

*Answer all questions using R-Script*
Question 1
Using the built in CO2 data frame, which contains data from an
experiment on the cold tolerance of Echinochloa crus-galli; find
the following.
a) Assign the uptake column in the
dataframe to an object called "x"
b) Calculate the range of x
c) Calculate the 28th percentile of
x
d) Calculate the sample median of
x
e) Calculate the sample mean of x and
assign it to an object called "xbar"
f) Calculate...

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