Question

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 logic and learnt to solve similar ones myself. I will rate your answer for you. thank you kindly!

Answer #1

(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...

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...

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.

Let V and W be vector spaces and let T:V→W be a linear
transformation. We say a linear transformation S:W→V is a left
inverse of T if ST=Iv, where ?v denotes the identity transformation
on V. We say a linear transformation S:W→V is a right inverse of ?
if ??=?w, where ?w denotes the identity transformation on W.
Finally, we say a linear transformation S:W→V is an inverse of ? if
it is both a left and right inverse of...

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.

Give an example of a linear transformation T:R2
-->R2 such that rank(T)=rank(T2) and T
does not equal T2. Write the matrix representation of
T(denoted [T]) with respect to the standard ordered basis

Let T be a 1-1 linear transformation from a vector space V to a
vector space W. If the vectors u,
v and w are linearly independent
in V, prove that T(u), T(v),
T(w) are linearly independent in W

Consider the transformation T: R2 -> R3 defined by
T(x,y) = (x-y,x+y,x+2y)
Answer the Following
a)Find the Standard Matrix A for the linear transformation
b)Find T([1
-2])
c) determine if c = [0 is in the range of the transformation
T
2
3]
Please explain as much as possible this is a test question that
I got no points on. Now studying for the final and trying to
understand past test questions.

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