Math 140A, Fall 2010, Quiz, 10/15/10
Instructions
. Answer all questions. You may use without proof anything which was
proved in class. If you need to cite a theorem, do so either by name, if it has one, or by
brieFy stating what it says.
1. (10 points) Let
a, b
be positive real numbers,
p
and
q
nonzero positive integers.
Explain what it means for
b
1
/p
=
a
. Using this, show that
(
b
1
/p
)
1
/q
=
b
1
/pq
.
b
1
/p
=
a
if
a
is the unique positive number such that
a
p
=
b
.
Note that (
a
q
)
p
=
a
pq
since
p, q
are integers, for
a
a positive real number. If
a
=
b
1
/pq
,
then
b
=
a
pq
= (
a
q
)
p
. Thus
a
q
=
b
1
/p
, so
a
= (
b
1
/p
)
1
/q
.
2. (10 points) Show that if
x
∈
R
k
and
x
·
y
= 0 for all
y
∈
R
k
, then
x
=
0
.
Suppose
x
n
= 0, say
x
= (
x
1
, . . ., x
k
) with some
x
i
n
= 0. Then let
y
be the vector all
of whose components are zero except the
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 Spring '08
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 Math

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