Math 136 Assignment #3
1. Let
A
be a 3
×
4 matrix, and
b
be a vector in
R
3
. Let
A
1
2
0

2
=
b
,
and
A
be row equivalent to the matrix
0
1
5

8
0
0
1

2
0
3
0
1
.
Find all solutions to
A
x
=
b
.
2. Let
v
1
=
1
2
3
,
v
2
=

1
1

2
,
v
3
=
3
3
8
and
v
4
=
2

4
1
.
(a) Determine whether or not the set
S
=
{
v
1
,
v
2
,
v
3
,
v
4
}
is linearly independent.
(b) Find a subset of
S
consisting of three linearly independent vectors.
3. Find all values of
a
for which the vectors
1

2
3
,
0
1
1
and
1
2
a

2

1
are linearly dependent.
4. Let
f
1
(
x
) =
x
2
+ 2
x
+ 3,
f
2
(
x
) =
x
2
+
x
+ 2,
f
3
(
x
) =
x
2
+ 3
x
+ 4 and
f
4
(
x
) =

x
2
+ 1. Write
one of these polynomials in terms of the others.
5. Let
A
=
a
b
c
d
, and
ad

bc
= 0. Prove that the columns of
A
are linearly independent.
6. Let
{
v
1
,
v
2
,
v
3
,
v
4
}
be a linearly independent set of vectors.
Determine whether or not the
following sets are linearly independent:
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 Spring '08
 All
 Math, Linear Algebra, Algebra, Vector Space, linearly independent vectors

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