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MAT067
University of California, Davis
Winter 2007
Homework Set 3: Exercises on Linear Spans and Bases
Directions
: Please work on all exercises! Hand in Problems 1 and 2 as your ”Calculational
Homework” and Problems 5 and 7 as your ”ProofWriting Homework” at the beginning of
lecture on January 26, 2007.
As usual, we are using
F
to denote either
R
or
C
.
1. Show that the vectors
v
1
= (1
,
1
,
1),
v
2
= (1
,
2
,
3), and
v
3
= (2
,

1
,
1) are linearly
independent in
R
3
. Write the vector
v
= (1
,

2
,
5) as a linear combination of
v
1
,
v
2
,
and
v
3
.
2. Consider the complex vector space
V
=
C
3
and the list (
v
1
, v
2
, v
3
) of vectors in
V
,
where
v
1
= (
i,
0
,
0),
v
2
= (
i,
1
,
0), and
v
3
= (
i, i,

1). Show that span(
v
1
, v
2
, v
3
) =
V
.
3. Find a basis for the subspace
U
of
R
5
deﬁned by
U
=
{
(
x
1
, x
2
, . . . , x
5
)

x
1
= 3
x
2
, x
3
= 7
x
4
}
.
4. Let
V
be a vector space over
F
, and suppose that the list (
v
1
, v
2
, . . . , v
n
) of vectors
spans
V
, where each
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 Fall '07
 Schilling
 Linear Algebra, Algebra

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