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110.201 Linear Algebra
3rd Quiz Solutions (Thursday)
March 24, 2005
Notation.
•
P
n
= space of polynomials, with real coeﬃcients, of degree at most
n
.
•
R
m
×
n
= space of
m
by
n
real matrices.
Problem 1
Determine whether the following spaces are isomorphic. In case
they are isomorphic, deﬁne an isomorphism relating them. Justify your answer.
Solution
Spaces are isomorphic if they have the same dimension.
1.
R
2
and
R
4
. No.
2.
P
5
and
R
5
No.
3.
R
2
×
3
and
R
6
Yes, under the natural identiﬁcation.
4.
P
5
and
R
2
×
3
Yes, under the natural identiﬁcation.
5.
R
2
×
k
and
C
k
, for
k
∈
N
. Yes, under the natural identiﬁcation.
Problem 2
Let
V
=
C
1
([0
,
1]) be the set of continuously diﬀerentiable func
tions on the closed interval [0
,
1].
V
is a real linear space with respect to the
operations of pointwise addition of functions and scalar multiplication.
(a) To prove that the functions
f
(
x
) = cos
x
,
g
(
x
) = 2
x
, and
h
(
x
) =
e
x
are
linearly independent in
V
, consider a relation
c
1
cos
x
+
c
2
2
x
+
c
3
e
x
= 0
.
Obviously this can only be true if
c
1
=
c
2
=
c
3
= 0.
(b) Given an integer
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 Spring '05
 CONSANI
 Linear Algebra, Algebra, Polynomials, Matrices

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