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Math 136
Assignment 7
Due: Wednesday, Mar 10th
1.
Show the each of the following sets form a basis for the subspace that they span, and
determine the coordinates of
±x
and
±
y
with respect to the basis.
a)
{
(1
,
1
,
0
,
1
,
0)
,
(1
,
0
,
2
,
1
,
1)
,
(0
,
0
,
1
,
1
,
3)
}
;
= (2
,

2
,
5
,

1
,

5),
±
y
=(

1
,

3
,
3
,

2
,

1).
b)
±²
11
10
³
,
²
01
³
,
²
20
0

1
³´
;
=
²
12
³
,
±
y
=
²

41
14
³
.
2.
Find a basis and determine the dimension of the following sets.
a)
S
= span
{
(1
,
2
,
1)
,
(2
,
3
,

2)
,
(

1
,
0
,
7)
,
(2
,
2
,
1)
}
.
b)
S
= span
{
1+
x
+
x
2
,x
+
x
2
+
x
3
,
x
2
+
x
3
}
.
3.
Find a basis for the hyperplane
x
1

x
2
+
x
3
+2
x
4
= 0 in
R
4
and then extend the basis
to obtain a basis for
R
4
.
4.
Find a basis for
R
3
that includes the vectors (1
,
2
,

1) and (3
,

1
,
1).
5.
Find a basis for
M
(2
,
2) that includes the vectors
±v
1
=
²
³
,±v
2
=
²
21
³
.
6.
Suppose that
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This note was uploaded on 04/30/2010 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.
 Winter '08
 All
 Sets

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