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Math 294 Homework 4
Section 3.3
28. Form a 4 X 4 matrix A with the given vectors as its columns. The matrix
A reduces to
1
0
0
2
0
1
0
3
0
0
1
4
0
0
0
k

29
.
This matrix can be reduced further to
I
4
if and only if
k
6
= 29. By
Summary 3.3.9, the four given vectors form a basis of
R
4
unless
k
= 29.
32. We need to ﬁnd all vectors
~x
in
R
4
such that
x
1
x
2
x
3
x
4
.
1
0

1
1
= 0 ;
x
1
x
2
x
3
x
4
.
0
1
2
3
= 0
.
Which amounts to ﬁnding the kernel of
±
1
0

1
1
0
1
2
3
²
.
Using Kyle Numbers, we ﬁnd the basis
1

2
1
0
,

1

3
0
1
.
46. Use Exercise 45 with
~
v
1
=
1
2
3
4
,
~
v
2
=
1
4
6
8
, and
~w
i
=
~
e
i
for
i
=
1
,
2
,
3
,
4.
Now rref
1
1
1
0
0
0
2
4
0
1
0
0
3
6
0
0
1
0
4
8
0
0
0
1
=
1
0
2
0
0

1
/
4
0
1

1
0
0
1
/
4
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