Math 136  Final Exam Spring 2009
NOTE:
The questions on this exam does not exactly refect which questions will be on this terms
exam. That is, some questions asked on this exam may not be asked on our exam and there may
be some questions on our exam not asked here.
1.
Short Answer Problems
a) What can you say about the consistency and number oF parameters in the general solution
oF a system oF 4 linear equations in 5 variables iF the rank oF the coe±cient matrix is 4.
b) Let
B
=
{
x
2
+1
,x
2
+2
x,x
}
be a basis For
P
2
. IF [
±v
]
B
=
1
2
3
, ²nd
.
c) Let
±x
= (1
,

1
,
2) and
±
y
=(

2
,
1
,
1). Calculate
×
±
y
and
·
±
y
.
d) List 3 things equivalent to an
n
×
n
matrix being invertible.
e) Let
±a
be any ²xed vector in
R
n
. Prove that proj
(perp
(
)) =
±
0 For all
∈
R
n
.
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
=0
a) Write the coe±cient matrix
A
and augmented matrix For the system.
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 Winter '08
 All
 Math, Linear Algebra, FINAL EXAM SPRING

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