Question

(12) (after 3.3)

(a) Find a linear transformation T : R2 → R2 such that T (x) =
Ax that reflects a

vector (x1, x2) about the x2-axis.

(b) Find a linear transformation S : R2 → R2 such that T(x) =
Bx that rotates a

vector (x1, x2) counterclockwise by 135 degrees.

(c) Find a linear transformation (with domain and codomain)
that has the effect

of first reflecting as in (a) and then rotating as in (b).
Give the matrix of this

transformation explicitly. How is this transformation related
to T and S?

(d) Find the matrix representing the linear transformation
that first rotates as in (b), then reflects as in (a), and then
rotates backwards (i.e., clockwise by 135

degrees).

(e) What matrix do you get if you repeat the sequence in part
(d) ten times?

Write this matrix in terms of A and B. Can you write this
matrix explicitly?

Answer #1

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...

3.) Find the linear transformation T : R2 to R2 described
geometrically by "first rotate counter-clockwise by 60 degrees,
then reflect across the line y = x, then scale vectors by a factor
of 5". Is this linear transformation invertible? If so, find the
matrix of the inverse transformation.

(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...

Find the matrix A in the linear transformation y =
Ax,where a point x = [x1,x2]^T is projected on the x2 axis.That
is,a point x = [x1,x2]^T is projected on to [0,x2]^T . Is A an
orthogonal matrix ?I any case,find the eigen values and eigen
vectors of A .

Problem 2. (20 pts.) show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x1, x2, ...
are not vectors but are entries in vectors. (a) T(x1, x2, x3, x4) =
(0, x1 + x2, x2 + x3, x3 + x4) (b) T(x1, x2, x3, x4) = 2x1 + 3x3 −
4x4 (T : R 4 → R)
Problem 3. (20 pts.) Which of the following statements are true
about the transformation matrix...

3. Find the linear transformation T : R2 → R2 described
geometrically by “first rotate coun- terclockwise by 60◦, then
reflect across the line y = x, then scale vectors by a factor of
5”. Is this linear transformation invertible? If so, find the
matrix of the inverse transformation.

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.

Assume that T is a linear Transformation.
a) Find the Standard matrix of T is T: R2 -> R3 first rotate
point through (pie)/2 radian (counterclock-wise) and then reflects
points through the horizontal x-axis
b) Use part a to find the image of point (1,1) under the
transformation T
Please explain as much as possible. This was a past test
question that I got no points on. I'm study for the final and am
trying to understand past test questions.

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.

Problem 2. Show that T is a linear transformation by finding a
matrix that implements the mapping. Note that x1,
x2, ... are not vectors but are entries in vectors.
(a) T(x1, x2, x3,
x4) = (0, x1 + x2, x2 +
x3, x3 + x4)
(b) T(x1, x2, x3,
x4) = 2x1 + 3x3 − 4x4
(T : R4 → R)
Please show T is a linear transformation for part (a) and
(b).

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