MAS 213: Linear Algebra II.
Problem list for Week #2.
Tutorial on 14th September.
This week’s topics:
•
The span of a set.
•
Linear dependence and independence.
Tutorial problems:
Problem 1:
(Problems 1.4.3(e), 1.4.4(c) in [FIS].)
In the following parts you are given a vector space
V
, a vector
v
in it, and
a subset
S
of the vector space.
Determine if the given vector
v
lies in the span of the given subset
S
. If
it does, ﬁnd an explicit representation of
v
as a linear combination of the
vectors in
S
.
1.
V
=
R
3
,
v
= (5
,
1
,

5), and
S
=
{
(1
,

2
,

3)
,
(

2
,
3
,

4)
}
.
2.
V
=
P
3
(
R
),
v
=

2
x
3

11
x
2
+ 3
x
+ 2,
and
S
=
{
x
3

2
x
2
+ 3
x

1
,
2
x
3
+
x
2
+ 3
x

2
}
.
3.
V
=
M
2
×
2
(
R
),
v
=
[
2

1
1
3
]
,
and
S
=
{[
1 1
1 1
]
,
[
0 1
1 1
]
,
[
0 0
1 1
]}
.
Problem 2:
(Problem 1.4.6 in [FIS].)
Consider the following subset of
R
3
:
S
=
{
(1
,
1
,
1)
,
(1
,
1
,
0)
,
(1
,
0
,
0)
}
. Prove
that span(
S
) =
R
3
.
1