Please provide answers! No work needed. High rating guaranteed! I will also give a big tip if finished by tonight!Darshan Sandhu
51
WeBWorK assignment due : 04/13/2016 at 03:00am EDT.
1.
(1 pt) Find the matrix
A
of the linear transformation
T
from
R
2
to
R
2
that rotates any vector through an angle of 150
?
in the counterclockwise direction.
A
=
±
²
.
2.
(1 pt) To every linear transformation
T
from
R
2
to
R
2
,
there is an associated 2
×
2 matrix. Match the following linear
transformations with their associated matrix.
1. Re?ection about the line y=x
2. Re?ection about the
x
axis
3. Counterclockwise rotation by
?
/
2 radians
4. Clockwise rotation by
?
/
2 radians
5. The projection onto the xaxis given by T(x,y)=(x,0)
6. Re?ection about the yaxis
A.
³
1
0
0
0
´
B.
³
0
1
1
0
´
C.
³

1
0
0
1
´
D.
³
1
0
0

1
´
E.
³
0
1

1
0
´
F.
³
0

1
1
0
´
G. None of the above
3.
(1 pt)
Let
T
:
R
2
?
R
2
be the linear transformation that ?rst rotates
points clockwise through 30
?
and then re?ects points through
the line
y
=
x
.
Find the standard matrix
A
for
T
.
A
=
±
²
.
4.
(1 pt) Find the characteristic polynomial of the matrix
A
=
?
?
4
3
0
0
5
5
2
2
0
?
?
.
p
(
x
) =
.
5.
(1 pt) If
v
1
=
±
5
3
²
and
v
2
=
±
5
4
²
are eigenvectors of a matrix
A
corresponding to the eigenvalues
?
1
=

2 and
?
2
=

1, respectively,
then
A
(
v
1
+
v
2
) =
±
²
and
A
(

3
v
1
) =
±
²
.
6.
(1 pt) Find a basis of the eigenspace associated with the
eigenvalue 1 of the matrix
A
=
?
?
?
?
1
0
4
0
2
1
3
1
1
0
3
0
1
0
3
0
?
?
?
?
.
?
?
?
?
?
?
?
?
,
?
?
?
?
?
?
?
?
7.
(1 pt) The matrix
A
=
?
?
6
6
12
3
3
6
3
3
6
?
?
has two real eigenvalues, one of multiplicity 1 and one of mul
tiplicity 2. Find the eigenvalues and a basis of each eigenspace.
?
1
=
has multiplicity 1,
Basis:
?
?
?
?
,
?
2
=
has multiplicity 2,
Basis:
?
?
?
?
,
?
?
?
?
.
8.
(1 pt) Let
A
=
±
53
36
72
49
²
.
Find an invertible
matrix
P
and a diagonal matrix
D
such that
PDP

1
=
A
.
P
=
±
²
D
=
±
²
9.
(1 pt) Let
A
=
?
?
3
0
0
0
3
0
12
12
1
?
?
.
Find an invert
ible matrix
P
and a diagonal matrix
D
such that
D
=
P

1
AP
.
P
=
?
?
?
?
D
=
?
?
?
?
10.
(1 pt) Let
M
=
±
1
2
4
7
²
.
Find formulas for the entries of
M
n
, where
n
is a positive inte
ger.
M
n
=
±
²
.
Generated by the WeBWorK system c
±
WeBWorK Team, Department of Mathematics, University of Rochester
1
Answer