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Problem Set 3
Math 115A/5 — Fall 2011
Due: Friday, October 21
Problem 1
Consider the function
T
:
P
(
F
)
→ F
(
F,F
) which takes a polynomial
f
(
x
) =
a
n
x
n
+
···
+
a
0
with coeﬃcients in
F
to the corresponding function
f
:
F
→
F
obtained by evaluation of
the polynomial.
(a)
Show that
T
is a linear transformation.
(b)
Show that, if
F
is inﬁnite,
T
is injective.
(c)
For
F
=
F
2
, the ﬁeld of 2 elements, ﬁnd a polynomial in the null space of
T
.
Problem 2
(1.6.20)
Let
V
be a vector space having dimension
n
, and let
S
be a subset of
V
that generates
V
.
(a)
Prove that there is a subset of
S
that is a basis for
V
.
(b)
Prove that
S
contains at least
n
vectors.
Problem 3
(1.6.33)
(a)
Let
W
1
and
W
2
be subspaces of a vector space
V
such that
V
=
W
1
⊕
W
2
. If
β
1
and
β
2
are bases for
W
1
and
W
2
, respectively, show that
β
1
∩
β
2
=
∅
and
β
1
∪
β
2
is a basis
for
V
.
(b)
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This note was uploaded on 04/09/2012 for the course MATH 172a taught by Professor Kong,l during the Fall '08 term at UCLA.
 Fall '08
 Kong,L
 Math

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