Math 417 homework 2 solutions
(Given only for problems that are not straightforward computation; contact the instructor if you still have questions about the others.)
Section 2.1 problem 1
A transformation T : R3 R3 is linear only if it satisfies
Math 417: Matrix Analysis (Section 1, Fall 2007)
Instructor
Name: Volker Elling Office: East Hall 4063 (after Thu Sep 6) Office hours: MWF 2:30-3:30 pm or by appointment; feel free to walk in any time Office phone: (734)764-0366 Uniqname: velling (fo
Learning Outcomes for Math 113
At the end of the course, Math 113 students should be able to.
1. compute dot products, projections, determinants, cross products, equations of planes and
volumes of parallelopipeds.
2. use the dot product to nd the angle be
Recipes for solving distance problems
These notes are short summaries of the procedure for solving distance problems.
Distance between two lines:
Let L1 be a line and P be a point on it. Let v1 be a vector parallel to L1 .
Let L2 be a line and Q be a poin
Second Midterm Exam Solutions
Math 21a
1
Spring, 2009
(10 points) Find all critical points of f (x, y) = x2 y x2 2y 2 , and classify each as a local minimum,
local maximum, or saddle point.
Solution: Recall that the critical points are where f = 0, or fx
Second Midterm Exam
Math 21a
Your Name
Spring, 2009
Your Signature
Instructions:
Please begin by printing and signing your name in
the boxes above and by checking your section in
the box to the right.
MWF 9
John Hall
MWF 10
Janet Chen
You are allowed 2
Math 21a
1
Second Old Final Exam
Spring, 2009
True-False questions. Circle the correct letter. No justications are required.
T F Any parameterized surface S is a graph of a function f (x, y).
T F If F is a vector eld of unit vectors dened in 1/2 x2 + y 2
First Old Final Exam
Math 21a
Spring, 2009
Multiple choice. Each problem has a unique correct answer. You do not need to justify your
answers in this part of the exam.
Part I:
1
True-False questions. Circle the correct letter. No justications are required
Math 21a
First Midterm Exam
Your Name
Your Signature
Spring, 2009
Instructions:
Please begin by printing and signing your name in
the boxes above and by checking your section in
the box to the right.
MWF 9
John Hall
MWF 10
Janet Chen
You are allowed 2 h
Math 21a
1
First Midterm Exam Solutions
Spring, 2009
(8 points)
(a) (4 points) Find an equation for the plane containing the three points P (3, 3, 1), Q(2, 1, 0),
and R(1, 3, 1).
Solution:
One normal for this plane is P Q P R = 1, 4, 1 4, 6, 0 =
6, 4, 10
Fourth Old Final Exam
Math 21a
1
Spring, 2009
True-False questions. Circle the correct letter. No justications are required.
T F The projection vector projv (w) is parallel to w.
T F If the directional derivatives Dv f (1, 1) and Dw f (1, 1) are both 0 fo
Final Exam Solutions
Math 21a
1
Spring, 2009
(5 points) Indicate whether the following statements are True or False by circling the appropriate
letter. No justications are required.
T F The (vector) projection of 3, 17, 19 onto 1, 2, 3 is equal to the (ve
Final Exam
Math 21a
Your Name
Spring, 2009
Your Signature
Instructions:
Please begin by printing and signing your name in
the boxes above and by checking your section in
the box to the right.
MWF 9
John Hall
MWF 10
Janet Chen
You are allowed 3 hours (18
Solutions to old Exam 1 problems
Hi students!
I am putting this old version of my review for the rst midterm
review, place and time to be announced. Check for updates on
the web site as to which sections of the book will actually be
covered. Enjoy!
Best,
Math 417 homework 1 solutions
1
Solutions
Most of the questions are straightforward computation; contact the instructor if you still have questions.
Section 1.3 problem 24
Let A R44 , b R4 . Assume that Ax = b has a unique solution x. If we cha
Math 417 homework 3 solutions
(Given only for problems that are not straightforward computation; contact the instructor if you still have questions about the others.)
Problem 12,20,24
Finding things "by inspection" is of course whichever way you ca
Math 417 homework 4 solutions
(Given only for problems that are not straightforward computation; contact the instructor if you still have questions about the others.)
Section 2.3 problem 4,5,20
To invert a matrix A, bring A to reduced row echelon f
Math 417 homework 5 solutions
Section 3.4 Problem 22
A= -3 4 , 4 3 S= 1 -2 2 1
(the columns of S are the basis vectors). S -1 can be computed either by elementary row operations, or by formula which is faster here: S -1 = Thus, by Fact 3.4.4, B = S
Math 417 homework 7 solutions
Section 6.1 problem 6
Matrix is triangular, so determinant is product of diagonal entries: 6 4 1 = 24. Nonzero, so matrix invertible.
Section 6.1 problem 8
The matrix does not have any zeros or other nice features, s
Math 417 homework 8 solutions
Problem 1
The matrix is cos sin Characteristic polynomial: (cos - z)2 + sin2 Roots: solve (cos - )2 + sin2 = 0. Get: (cos - )2 = - sin2 , so cos - = i sin , so = cos i sin . - sin . cos
Problem 2
Result:
Math 417 homework 9 solutions
Problem 1
Characteristic polynomial: -3 - z A (z) = det(A - zI) = det 2 0 -2 -3 - z 0 0 0 4-z
= (-3 - z)2 - 2(-2) (4 - z) = (3 + z)2 + 4)(4 - z). One root: z = 4. The other two are obtained by solving 0 = (3 + z)2
Math 417 homework 10 Solutions
Problem 1
(a) Show: if v1 , v2 are eigenvectors of a matrix A for the same eigenvalue , then their linear combinations 1 v1 + 2 v2 are also eigenvectors (except for those = 0) for that eigenvalue . Answer: A(1 v1 + 2 v