Question

find the standard matrix for the stated composition R^3 . A
rotation of 30 about the x-axis followed by a rotation of 30 about
z-axis

Answer #1

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Simple Rotation Matrices
1) Derive the rotation matrix for simple rotation about the
x-axis
•2) Derive the rotation matrix for simple rotation about the
y-axis
•3) Derive the rotation matrix for simple rotation about the
z-axis
•Hint: Three different methods are covered in the
supplementary
notes. Please study them and use each method only once.

Axis of Rotation (3d)
The following matrix is the matrix of rotation. Find the axis of
rotation.
-6/11
6/11
-7/11
-2/11
-9/11
-6/11
-9/11
-2/11
6/11

(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

Find the standard matrix for the following transformation T : R
4 → R 3 : T(x1, x2, x3, x4) = (x1 − x2 + x3 − 3x4, x1 − x2 + 2x3 +
4x4, 2x1 − 2x2 + x3 + 5x4) (a) Compute T(~e1), T(~e2), T(~e3), and
T(~e4). (b) Find an equation in vector form for the set of vectors
~x ∈ R 4 such that T(~x) = (−1, −4, 1). (c) What is the range of
T?

a. Let →u = (x, y, z) ∈ R^3 and define T : R^3 → R^3 as
T( →u ) = T(x, y, z) = (x + y, 2z − y, x − z)
Find the standard matrix for T and decide whether the map T is
invertible.
If yes then find the inverse transformation, if no, then explain
why.
b. Let (x, y, z) ∈ R^3 be given T : R^3 → R^2 by T(x, y, z) = (x...

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.

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 ?.

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?

. 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 =...

Find Eigenvalues and Eigenspaces for matrix:
The 2 × 2 matrix AT associated to the linear transformation T :
R2 → R2 which rotates a vector π/4-radians then reflects it about
the x-axis.

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