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Math 21b Final Exam  Spring 2000
This was a three hour exam. Total points = 90.
(1)
(12 pts) Let
1 234
5 6
78
9
1
0
1
1
12

=
A
.
(a) (2 pts) Find rref(
A
), the reduced row echelon form of
A
.
(b) (4 pts) Find bases for ker(
A
) and image(
A
).
(c) (6 pts) Find an orthonormal basis for ker(
A
).
Now let
0
2
4
2
=
v
. Show that
v
∈
ker(
A
) and express
v
in terms of your orthonormal basis for ker(
A
).
(2)
(10 pts) Let
0 001
0 010
0 100
1 000
=


J
.
We say that a 4
×
4 matrix
A
is symplectic if
AJA
T
=
J
.
(a) (2 pts) Show that if A is symplectic, then
A
1
is
symplectic.
(b) (2 pts) Show that if
A
and
B
are symplectic, then
AB
is symplectic.
(c) (2 pts) Show that
J
itself is symplectic.
(d) (4 pts) Given
v
,
w
∈
R
4
, define
<
v
,
w
=
v
T
Jw
. Is this
an inner product on
R
4
? Why or why not?
(3)
(10 pts) Find all solutions to the differential equation
3
(
)2 (
)
(
)4
t
f
t
f
t
f
te
′
′′
 +=
.
Find the unique solution given the initial conditions
(0
)1
f
=
and
(0
f
′
=
.
(4)
(8 pts) Let
V
= {
B
∈
M
n
(
R
):
B
+
B
T
=
0
} be a set of real
n
×
n matrices. (Recall that such matrices are called
antisymmetric
.) Show that
V
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