1

1
F
:
R
3
!
R
3
such that ker(
F
) =
R
(
F
).
(b) Find a linear transformation
F
:
R
2
!
R
2
such that ker(
F
) =
R
(
F
). You must explicitly
show that your example has this property.
2
:
Let
F
:
R
3
!
R
2
be the linear transformation
F
(
x;y;z
) = (
x;y
+
z
) and let
G
:
R
2
!
R
3
be
dened by
G
(1
;
0) = (0
;
1
;
2) and
G
(0
;
1) = (0
;
2
;
0).
(a) Find the standard matrix of the composition
G
F
.
(b) Find the kernel and range of
G
F
.
(c) Find the standard matrix of the composition
F
G
.
(d) Find the kernel and range of
F
G
.
3
:
Let
F
:
R
2
!
R
2
be (anticlockwise) rotation by
3
4
, let
G
:
R
2
!
R
2
be the orthogonal projection
onto the line
y
=
3
x
, and let
H
:
R
2
!
R
2
be a dilation by a factor of
p
8 (i.e. a dilation with
t
=
p
8). Find the standard matrix for each of the following linear transformations:
(a)
F
H
G
(b)
F
G
H
(c)
H
G
F
.
4
:
(a) Find the standard matrix
A
of the linear transformation
F
representing the cyclic permu-
tation
F
(
x
1
;x
2
;x
3
;x
4
) = (
x
4
;x
1
;x
2
;x
3
)
:
(b) What is the eect of the composition
F
F
? What is the standard matrix for
F
F
?
(c) Show that
A
3
=
A
1
. Why would you have expected this for the standard matrix of
F
?
(d) Is
F
invertible? Explain.
5
:
Let
F
:
R
l
!
R
m
and
G
:
R
m
!
R
n
be linear transformations.
(a) Show that ker(
F
)
ker(
G
F
).
(b) Show that
R
(
G
F
)
R
(
G
).
6
:
Classify the following conic sections by completing the square to put them into standard form
and sketch each one:
(a) 4
x
2
+
y
2
20
x
+ 24 = 0
(b)
x
2
+
y
2
2
x
+ 2
y
+ 2 = 0
(c)
y
2
+ 3
x
+ 6
y
+ 8 = 0
(d)
4
x
2
+ 9
y
2
4
x
6
y
+ 3 = 0

2