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1.
a) If the rank is 4 and there are 4 linear equations, then there is a leading 1 in each
row of the coeFcient matrix and hence the system is consistent for all
±
b
. The number of
parameters in the general solution is 54=1.
b)
±v
= 1(
x
2
+ 1) + 2(
x
2
+2
x
) + 3(
x
+ 1) = 3
x
2
+7
x
+ 4.
c)
±x
·
±
y
= 1(

2) + (

1)(1) + 2(1) =

1,
×
±
y
=(

3
,

5
,

1).
d) There are many, many possible choices.
e)
proj
±a
(perp
(
)) =
·
(

·
±
±
2
)
±
±
2
=
·

·
±
±
2
±
±
2
±
±
2
=
·

·
±
±
2
=0
2.
Consider the system of linear equations
x
1
x
2
+

x
4
=2

x
1

2
x
2
+
x
3
+4
x
4
=

4
2
x
1
x
2
x
3
x
4
a)
1
2
0

1

1

2 1
4
2
4
2
4
1 2 0

1
2
0 0 1
3

2
0 0 2
6

4
b)
x
1
x
2
x
3
x
4
=
2
0

2
0
+

2
1
0
0
s
+
1
0

3
1
t.
c) ±rom our work in b) we have
A
=
1
0 0

1 1 0
0
0 1
1 0 0
0 1 0
2 0 1
1 0 0
0 1 0
21
1 2 0

1
0 0 1
3
0 0 0
0
.
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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
 Linear Equations, Equations

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