Final Exam
Fall 2009
Name: _ This is a closed book exam. You may use a calculator and the formulas handed out
with this exam. You may find that your calculator can do some of the problems. If this is so, you still need to show
how to do the problem by han
Assignment 8
Due:
Monday, March 14. Nothing accepted after Tuesday, March 15. 10% off for being late.
Please work by yourself on this assignment and all the other assignments that you turn in as part of
your grade. Please see me if you need help.
Let B =
Exam 1
Math 217
Winter 2010
Name: _ This is a closed book exam. You may use a calculator and the formulas handed out
with this exam. You may find that your calculator can do some of the problems. If this is so, you still need to show
how to do the problem
Exam 3
Fall 2009
Name: _ This is a closed book exam. You may use a calculator and the formulas
handed out with this exam. You may find that your calculator can do some of the problems. If this is so,
you still need to show how to do the problem by hand, e
Exam 1
Fall 2009
Name: _ This is a closed book exam. You may use a calculator and the formulas handed out
with this exam. You may find that your calculator can do some of the problems. If this is so, you still need to show
how to do the problem by hand, e
10 Eigenvalues of Multiplicity Greater Than One
So far we have restricted out attention to matrices whose eigenvalues have multiplicity
one. In this chapter we consider matrices which have eigenvalues of multiplicity greater
than one. Suppose A is a matri
Assignment 2
Due:
Monday, January 24. Nothing accepted after Thursday, January 27. 10% off for
being late.
Please work by yourself on this assignment and all the other assignments that you
turn in as part of your grade. Please see me if you need help.
1.
Assignment 3
Due:
Monday, January 31. Nothing accepted after Tuesday, February 1. 10% off for
being late.
Please work by yourself on this assignment and all the other assignments that you
turn in as part of your grade. Please see me if you need help.
1. a
Exam 2
Math 217
Winter 2010
Name: _. This is a closed book exam. You may use the formula sheet handed out
with the exam. Show all work and explain any reasoning that is not clear from the computations. Turn in
this exam with your answers.
1.
2.
3.
4.
5.
(
Exam 2
Fall 2009
Name: _ This is a closed book exam. You may use a calculator and the
formulas handed out with this exam. You may find that your calculator can do some of the
problems. If this is so, you still need to show how to do the problem by hand, e
10.2 Generalized Eigenvectors
In the previous section we looked at the case where each eigenvalue of a square matrix A
has as many linearly independent eigenvectors as its multiplicity. In that case we could
diagonalize A and use this to compute its power
VJS
VA
J
>
\*
3
\
 <tl
l
^
K
(N
i
,
*>
\
M
(\
0
o
M
i
f
^ o
r\
5.
15
r
VI
H
t
i\
O
^
^
rv
in
?
1
O
M
1
D
AJ K ^ A
/o

*
/A<
ma4riY
bf
G>
O
A
Cotsnkr 
c(ty(,l^t^,
r&ld'^
\oi*
Crvi
Artjif
Q
~<?'si
c
J
10*
2_
i
^
, H t*</ ]
<*
I
D
<>
q
<v
<a
0
 J
z
Formulas
Trigonometric Identities
sin(x+y) = sin x cos y + cos x sin y
cos(x+y) = cos x cos y  sin x sin y
sin x sin y = [ cos(xy)  cos(x+y) ]
cos x cos y = [ cos(xy) + cos(x+y) ]
sin x cos y = [ sin(x+y) + sin(xy) ]
sin2x = [ 1  cos(2x) ]
cos2x = [
Final Exam
Math 217
Winter 2010
Name: _ This is a closed book exam. You may use the formula sheet handed out with the exam.
Show all work and explain any reasoning that is not clear from the computations. Turn in this exam with your
answers.
1.
A square m
Exam 3
Math 217
Winter 2010
Name: _ This is a closed book exam. You may use the formula sheet handed
out with the exam. Show all work and explain any reasoning that is not clear from the computations.
Turn in this exam with your answers.
1.
Let v1 = , v2
LIMITS AT INFINITY
 In this section, we consider limits of functions where the
independent variable either increases or decreases without
bound.
Consider:
 The statement
lim () =
+
means that the value of () approaches L as the value of
increases with