Math 601
HOMEWORK 1
1.
Decide which of the following sets are subspaces in
R
3
, and for each
subspace ﬁnd a basis, its dimension and describe the type of geometrical
ﬁgure it is.
U
1
=
{
(
x
1
,x
2
,x
3
)

x
1
= 2
t, x
2
=

t, x
3
= 5
t, t
∈
R
}
U
2
=
{
(
x
1
,x
2
,x
3
)

x
1
= 2
t
+ 1
, x
2
=

t, x
3
= 5
t, t
∈
R
}
U
3
=
{
(
x
1
,x
2
,x
3
)

x
1
= 2
t
2
, x
2
=

t, x
3
= 5
t, t
∈
R
}
U
4
=
{
(
x
1
,x
2
,x
3
)

x
1
= 2
t, x
2
=

t, x
3
= 5
t, t >
0
}
U
5
=
Sp
(
x,y,z
) with
x
= (1
,
1
,
0)
, y
= (

2
,

2
,
0)
, z
= (0
,
1
,
1)
2.
Show that the intersection
U
∩
V
of two subspaces
U, V
is also a
subspace.
3.
Show that the sum
U
+
V
of two subspaces
U, V
is also a subspace.
4.
Consider the vector space of polynomials of degree at most
n
, with
coeﬃcients in
R
:
P
n
=
{
p
(
t
) =
a
0
+
a
1
t
+
a
2
t
2
+
...
+
a
n
t
n

a
j
∈
R
}
Show that the monomials 1
, t,t
2
,... t
n
form a basis for
P
n
. Show that
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 Winter '08
 un
 Linear Algebra, Algebra, Vector Space, Sets, Complex number

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