MATH 235
Review of Linear Algebra I
Assignment 1
Hand in questions 2,4,7,8,9,12 by 9:30 am on Wednesday September 20, 2006.
1. Solve the following system of equations for
x
and
y
:
x
2
+
xy

y
2
= 1
2
x
2

xy
+ 3
y
2
= 13
x
2
+ 3
xy
+ 2
y
2
= 0
2. Solve
A
x
=
b
and
y
A
=
c
where
A
=
1

1

1
2

1

3
1
0

2
,
x
=
x
1
x
2
x
3
,
b
=
2
6
4
,
y
= (
y
1
, y
2
, y
3
)
,
c
= (0
,

2
,
2)
.
3. A system of linear equations in
x
1
, x
2
, x
3
has the augmented matrix
1
0
1
b
1
1
0
a
1
a
b
a
,
where
a
and
b
are real numbers. Determine values of
a
and
b
, if they exist, such that
the system has:
(a) no solution;
(b) a unique solution;
(c) infinitely many solutions.
For the values of
a
and
b
that you find in part (c) above, give the general solution to
the system of equations.
4. Find
A

1
for
A
=
1
i
2
2
0
6

i
1

i
.
5. Determine if the following set of vectors in
C
3
is linearly independent:
v
1
= (1
,
0
,

i
)
,
v
2
= (1 +
i,
1
,
1

2
i
)
,
v
3
= (0
, i,
2).
6. Determine whether the following subsets
W
are subspaces of
V
:
(a)
V
=
M
33
, W
=
{
A
∈
M
33
:
A
is nonsingular
}
;
(b)
V
=
M
nn
, W
=
{
A
∈
M
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 Fall '08
 CELMIN
 Math, Linear Algebra, Algebra, Equations, Vector Space, Linear map

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