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SAMPLE FINAL 2 ANSWERS
NOTE: These are only answers to the problems and not full solutions! On the ﬁnal exam
you will be expected to show all steps used to obtain your answer.
1.
a) If the rank is 4 and there are 4 linear equations, then there is a leading 1 in each
row of the coeﬃcient 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,
~x
×
~
y
=

3

5

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

~x
·
~a
k
~a
k
2
~a
)
k
~a
k
2
=
~a
·
~x

~a
·
~x
k
~a
k
2
k
~a
k
2
k
~a
k
2
=
~a
·
~x

~a
·
~a
k
~a
k
2
= 0
2.
Consider the system of linear equations
x
1
+ 2
x
2
+

x
4
= 2

x
1

2
x
2
+
x
3
+ 4
x
4
=

4
2
x
1
+ 4
x
2
+ 2
x
3
+ 4
x
4
= 0
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
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This note was uploaded on 12/22/2010 for the course MATH 136 taught by Professor All during the Fall '08 term at Waterloo.
 Fall '08
 All
 Linear Equations, Equations

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