Homework 3
Andrei Antonenko
Due at the beginning of the class, October 17, 2001
1. Which of the following statements are true or false? Explain your answer. If false, give a
counterexample.
(a) For any
u
1
,
u
2
, and
u
3
from
V
, the system
0
,u
1
,u
2
,u
3
is linearly dependent.
(b) If
u
1
,
u
2
, and
u
3
span
V
, then
u
1
,
u
2
,
u
3
, and
w
span
V
for any
w
∈
V
.
(c) If
u
1
,
u
2
, and
u
3
span
V
, then dim
V
= 3.
(d) If
u
1
,
u
2
, and
u
3
span
V
, then they are independent.
(e) If
u
1
,
u
2
, and
u
3
form a basis for
V
, then they are independent.
(f) If
u
1
,
u
2
are independent, then they form a basis of
V
.
(g) If
u
1
,
u
2
,
u
3
, and
u
4
are linearly independent, then dim
V
= 4.
(h) If
u
1
,
u
2
, and
u
3
are linearly independent, then dim
V
≥
3.
2. Consider the vector space
R
4
. Let
u
1
= (1
,
2
,

1
,
0),
u
2
= (2
,
1
,
1
,
1), and
u
3
= (

1
,
0
,
0
,
1).
(a) Find 2
u
1
+
u
2

u
3
.
(b) Find vector
x
∈
R
4
such that 2
u
1
+
u
2
+ 2
u
3
+
x
=
0
.
3. Consider the vector space
R
3
. Let
u
1
= (1
,
2
,
1),
u
2
= (2
,
1
,
3), and
v
= (1
,
2
,λ
)
, λ
∈
R
. Find
all
λ
’s such that the vector
v
can be expressed as a linear combination of
u
i
’s.
4. Determine which of the following sets with standard operations form a vector space (over the
ﬁeld
R
). Explain your answers.
(a)
{
(
x,y,x
)

x,y
∈
R
}
(b)
{
(0
,x,y
)

x,y
∈
R
}
(c)
{
(
x,y,
1)

x,y
∈
R
}
(d)
{
(
x,y,x
+
y
)

x,y
∈
R
}
(e)
{
(
x,y
)

x,y
∈
R
,x
≥
0
}
(f) Set of all polynomials of even power:
'
f
(
t
) =
∑
n
i
=0
a
i
t
i

a
i
∈
R
,n
is even
“
(g) Set of all symmetrical 2