UPenn Math Dept's Math 240 page ~ Math 240 newsgroup

Math 240,

the course of Dr. Mihailovs

Texts
Lectures
Mo Tu We Th 1:00--3:10 p.m. (June 29 - August 7) in DRL 4C2. Attendance contributes 10 % to total grade.
Homework
Must be handed in on Tu and Th. Expect to do about 15 hours of homework per week. Contributes 20 % to total grade.
Exams

Homework 1

(hand in on Tu, June 30)

Zill/Cullen

Maple manual

Homework 2

(hand in on Th, July 2)

Zill/Cullen

Maple manual Dr. Shale's problems
1.   Let A = (
0 -1
1 0
). Write down I, A, A2, ...   Hence find eAt.   Show eAt = (
cos t -sin t
sin t cos t
).  
2.   Let A = (
a 0
0 b
). Find eAt.
3.   Let A = (
a 1
0 a
). Find eAt.
4.  
Let e1 =
(
1
0
)
, e2 =
(
0
1
)
.
Let A be a 2×2 matrix. Then A determines a mapping of the plane onto itself which we denote by TA. For each A given below mark in the images under TA of each of the vectors 0, e1, e1 + e2, e2. By connecting the image points sketch the image of the oriented unit square D under TA. Say if TA preserves or reverses the orientation of D.
(i)   A = (
1 0
0 2
) ;     (ii)   A = (
0 1
1 0
) ;     (iii)   A = (
1 2
0 1
) .
5.  
For each 2×2 matrix A below, introduce x =
(
x1
x2
)
,   y =
(
y1
y2
)
and solve the system Ax = y for x. Next write your solution as x = By where B is a 2×2 matrix. Then B = A-1.
(i)   A = (
2 1
0 1
) ;     (ii)   A = (
2 0
0 -2
) ;     (iii)   A = (
1 2
3 4
) .
But the method fails for A = (
1 0
0 0
) which has no inverse. What does TA do in this case to D?
6.  
(i)   Is the following mapping from R2 to R2 linear,   T (
x
y
) = (
x + 1
y
) ?
(ii)   Let e be any unit vector in R3. Consider the mappings from R3 to R3,   P(x) = (x.e)e,   Q(x) = x - (x.e)e . Are these mappings linear?
(iii)   Because we can add them and multiply them by numbers, and all the requirements work out, functions on the line form an (infinite dimensional) vector space. For this space, which of the following mappings are linear?
a)   Tf(x) = f(x) + 1 ;  b)   Tf(x) = (f(x))2 ;  c)   Tf(x) = x f(x) ;  d)   Tf(x) = f'(x) . 
Handout

Homework 3

(hand in on Tu, July 7)

Maple manual

Dr. Shale's handout (13 problems)

Homework 4 - 10

Handout

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