Another method is to consider
T
c
=
B
1
(
0
)
∪{
x
:
k
x
k
>
1
}
and show it is open. Since the open
ball of radius 1 is open and the union of open sets is open, this reduces to showing
{
x
:
k
x
k
>
1
}
is open.
4. Consider the following set of vectors in
R
3
:
B
=
1
1
1
,
1
2
1
,
1
1
2
,
2
1
2
.
a
) Is
B
a linearly independent set? Justify your answer.
Answer:
No. No more than three vectors can be linearly independent in
R
3
.
b
) Does
B
span
R
3
? Justify your answer.
Answer:
Yes, it spans
R
3
. In fact, the first three elements of
B
span
R
3
. This can be seen by
calculating
1
1
1
1
2
1
1
1
2
= 1
.
Since the determinant is nonzero, the three vectors span
R
3
.
c
) Is
B
a basis for
R
3
?
Answer:
No. It is not a linearly independent set.
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 Spring '08
 STAFF
 Economics, Linear Algebra, Vector Space, Metric space, α, 1 w, 1 2w

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