Michael Bowen's VC Course Pages

# Math V22, Fall 2016

## Introduction and Announcements

Welcome to Math V22 (Introduction to Linear Algebra) at Ventura College. Michael Bowen (email) will be teaching this course during the fall 2016 term.

Important note: This web page is not a substitute for attending class; regular attendance is an expectation of this course. Modifications to homework assignments, and other important news announced in class, may not appear on this page for several days. You are still responsible for all assignments and in-class announcements even if they do not appear here! If you wish to verify information on this page, please contact the instructor.

## Textbook Information

The ISBN number is provided as a convenience if you wish to purchase this item online. The VC bookstore may stock a different ISBN number; either may be used for the course. If you buy from the bookstore, obtain the least expensive version you can find; do not pay extra for MyMathLab, WebAssign, or other software. Exception: You may wish to purchase the textbook bundled with the student edition of Maple, which is only available from the bookstore, not online. Some of the homework assignments will ask you to use Maple software (either your own copy, or a copy installed in the BEACH computer lab located on the first floor of the LRC building). If you obtain the book from another source, please be sure to obtain the correct edition, as noted below. Older editions are, of course, much less expensive, but the homework problems are different.

Select any one of the following required texts:

• Author: H. Anton
• Title: Elementary Linear Algebra, Eleventh Edition
• ISBN-13: 978-1118473504 (this is the ISBN stocked at the VC bookstore; it is also available online)
• Author: H. Anton
• Title: Elementary Linear Algebra, Loose Leaf Eleventh Edition
• ISBN-13: 978-1118677308 (less expensive UNBOUND (loose pages) version; binder-ready)
• Author: H. Anton
• Title: Elementary Linear Algebra Applications Version, Eleventh Edition
• ISBN-13: 978-1118434413 (acceptable alternative version which contains an extra chapter of applications at the back)
• Author: H. Anton
• Title: Elementary Linear Algebra Applications Version, Loose Leaf Eleventh Edition
• ISBN-13: 978-1118474228 (less expensive UNBOUND (loose pages) version; binder-ready)

If you use the Kindle or other digital version, you will want to be able to refer to the information during class, so be sure you own a compatible device (laptop, mobile phone, or e-reader) that you are willing and able to bring to class with you every day.

These additional texts are optional; select none, any, or all. (Note that when there is a conflict between the solution given in the textbook and the solution given in the student solutions manuals, the textbook is usually correct.)

• Author: H. Anton
• Title: Student Solutions Manual to accompany Elementary Linear Algebra, Applications version, Eleventh Edition
• ISBN-13: 978-1118464427

## Holidays

Classes at Ventura College will meet Monday through Thursday each week of the term, excepting only the dates listed below.

• Monday 5 September 2016 (Labor Day)
• Friday 11 November 2016 (Veterans' Day)
• Thursday 24 November–Friday 25 November 2016 (Thanksgiving Holidays)

Please note that Columbus Day and Halloween are not Ventura College holidays.

## Homework Club (Office Hours) During Finals Period

• Monday 5 December 2016: 1:00 to 2:00 p.m. in the Tutorial Center
• Tuesday 6 December 2016: 1:00 to 2:00 p.m. in the Tutorial Center
• Wednesday 7 December 2016: 1:00 to 2:00 p.m. in the Tutorial Center
• Wednesday 7 December 2016: 4:30 to 5:30 p.m. in SCI-223
• Monday 12 December 2016: 10:15 a.m. to 12:15 p.m. in SCI-352 (may move to MCE-227 if there is a conflicting final)
• You may also contact me by email; you may expect a response within 24 hours.

## Final Examination

Place/date/time:  Room SCI-354, Tuesday 13 December 2016, 7:15 p.m.

Important Note: This is 15 minutes later than our usual class meeting time!

Be sure that your big party to celebrate the end of finals occurs after the appropriate date. Requests for administration of early or late finals that require the instructor to reschedule his work or make a special trip to campus are subject to a deduction of points, regardless of the reason for the request.

## Homework Assignments

• These are listed in chronological order.
• Note: "EOO" (every other odd) means to do problems
1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, etc.
• Note: Please do "E.C." (extra credit) problems on a separate sheet of paper from the regular assignment. These are due at the next exam.
• These are tentative; the instructor may make changes to this list from time to time. Due dates will be announced in class.
• Students must not rely on printed versions of this list; instead, they should check the live online version periodically for possible updates.
• Students who work ahead and complete one or more assignments in advance are taking a risk that the assignment(s) may change before the due date, in which case the advice in the preceding sentence is particularly applicable.
• Students should not expect to earn extra credit for completing tentatively assigned exercises that are later modified or removed from this list.
Overview: This table lists homework assignments and announces examinations. It contains four columns. First row: Column headers. Second and subsequent rows: The homework due for each section covered in the course. Details: The first row is the table header, with column headings describing the data listed in the main body of the table. The second and subsequent rows contain section numbers and titles, assigned problem numbers, and extra credit problems, if any. Column one of these rows contains a section number. Column two of these rows contains the corresponding section title. Column three lists the problem numbers for each section. Column four lists extra credit problems, if any.
§ Title Problems and Supplements E.C.
Syllabus Worksheet (obtain a copy) (NOTE: This assignment is worth 15 points.)
1.1 Introduction to Systems of Linear Equations 1–15 ODD; 27 16
1.1 Lecture Notes
1.2 Gaussian Elimination 1; 3 (note that these are augmented matrices, with the rightmost column being the vector of constants); 11; 7 (it's easier to do #11 before #7 because the answer to #11 gets you halfway to the answer for #7; beware that the last equation has an $x$ term also, not just a $w$ term); 13–14 ALL (use theorem 1.2.2); 17–21 ODD (if there are multiple solutions, state them using parametric equations, as we did in lecture); 37; 43(a) Using Maple on your home/laptop computer (if installed) or in the BEACH computer lab, use the information on this page to enter the matrix corresponding to the system in problem 22; then use either the ReducedRowEchelonForm or GaussianElimination functions to help you solve the system; print and turn in your setup and results (some steps at the end will still need to be done by hand, so just write these in pen or pencil on the bottom of the printout)
1.2 Lecture Notes
Optional video supplement: MIT Open Courseware Lecture "Elimination with Matrices"
"Systems of Linear Equations" from the open-source textbook A First Course in Linear Algebra by R.A. Beezer
1.3 Matrices and Matrix Operations 1; 3; 5; 7; 11; 17; 21; 33
1.3 Lecture Notes
Online Matrix Calculator (found by Max Zielsdorf)
1.4 Inverses; Algebraic Properties of Matrices 1; 2; 3; 5; 13 (use the matrix in Exercise 5 for $A$, Exercise 6 for $B$, and Exercise 7 for $C$); 19; 21; 25; 39 (see solutions for related problems 40 and 50); 43 (multiply both sides by a cleverly chosen matrix); 46; 49 Any one proof taken from problems 44; 45; 47; or 51–58
1.4 Lecture Notes
1.5 Elementary Matrices and a Method for Finding $A^{-1}$ 1; 3; 11(a)(b); 13; 17; 23 (see solution for related problem 24; the back of the textbook provides one solution, but there are many other correct solutions, so use Maple to multiply your elementary matrices to verify your answer); 27; 32
1.5 Lecture Notes
1.6 More on Linear Systems and Invertible Matrices 1; 5; 10 (see Example 2 and the solution); 15 (the coefficient matrix is not invertible); 19 (first find the inverse of the $3 \times 3$ matrix on the left side of the equation)
1.6 Lecture Notes
1.7 Diagonal, Triangular, and Symmetric Matrices 1(a)(b)(c)(d); 3; 4; 10; 20; 21; 31; 37(a)(c); 44
1.7 Lecture Notes
1.8 Matrix Transformations 1(a)(b)(c)(d); 3(a)(b); 5(a)(b); 7(a)(b); 11(a)(b); 13(b)(d); 19(a)(b); 23(a)(b); 26; 27 32(a)
1.9 Applications of Linear Systems 5 (see solution for related problem 7); 11; 15; 17(a) T2 (in the Working with Technology section)
1.10 Leontief Input-Output Models (No required assignments; extra credit only) 4
Chapter Test 1

(Chapter 1)

Recommended study-guide problems

(These are study-guide problems for practice purposes; do not turn in with your homework)
(For students with minimal study time)
Page 101 (Supplementary Exercises): 1–10 ALL; 12; 14(a)(b); 15; 18; 19
(For students with additional study time) The above plus some or all of
Page 101 (Supplementary Exercises): 24
Section 1.8 (Exercise Set): 2(a)(b)(c)(d); 4(a)(b); 6(a)(b); 8(a)(b); 12(a)(b); 14(a)(b)(c)(d); 20(a)(b); 24(a)(b); 28
Additional problems taken from the unassigned homework exercises from chapter 1
2.1 Determinants by Cofactor Expansion 3; 5; 11; 13; 15; 18; 21; 23; 25; 29; 31; 33; 39 34 ("commute" means that $AB = BA$)
2.2 Evaluating Determinants by Row Reduction 5; 7; 9; 11; 15; 17; 19; 21; 25; 31
2.3 Properties of Determinants: Cramer's Rule 3; 5; 9; 13; 17; 21; 25; 27; 30; 35(a)(b)(c)(d)
3.1 Vectors in 2-Space, 3-Space, and n-Space 5; 7; 11(a)(b)(c)(d); 15; 17; 21; 23; 27
3.2 Norm, Dot Product, and Distance in ${\mathbb{R}^n}$ 1(a)(b); 7; 9(a)(b); 11(a)(b); 13; 15(a)(b)(c)(d); 17(a)(b); 27
3.3 Orthogonality 1; 3; 5; 7; 9; (if the planes are parallel, then their normal vectors will also be parallel; see page  156 of the textbook for a discussion of the point-normal equation of a plane); 11 (if the planes are perpendicular, then their normal vectors will also be perpendicular); 13(a)(b); 15; 17; 19; 29; 31; 33 24; 26; 28
3.4 The Geometry of Linear Systems 1–15 ODD 19; 23; 27
3.5 Cross Product 7; 9; 11; 13; 15; 17; 21; 25(a)(b)(c); 27(a); 28; 29
Chapter Test 2

(Chapters 2 and 3)

Recommended study-guide problems

(These are sample exam-like problems for practice purposes; do not turn in with your homework)
(For students with minimal study time)
Page 129 (Supplementary Exercises): 3; 5; 7; 9 (for #3 and #5; cross-check with your earlier results); 11 (for #1 and #3); 12 (for #5 and #7); 15; 16; 17–23 ODD; 26; 27; 29(a)
Page 181 (Supplementary Exercises): 1; 4(a)(b)(c)(d) (justify or draw illustrative diagrams); 7; 9–12 ALL; 13–21 ODD; 25
(For students with additional study time) The above plus some or all of
Page 129 (Supplementary Exercises): 4; 6; 8; 9 (for #4 and #6; cross-check with your earlier results); 11 (for #2 and #4); 12 (for #6 and #8)
Page 181 (Supplementary Exercises): 14–20 EVEN
Additional problems taken from the unassigned homework exercises from chapter 2
Additional problems taken from the unassigned homework exercises from chapter 3
4.1 Real Vector Spaces 3–11 ODD; 17 (to prove that a line not passing through the origin is not a vector space, all you have to do is prove that any one of the ten vector spaces axioms is not satisfied) 24
4.2 Subspaces 1; 3; 7; 9; 11; 13 14
4.3 Linear Independence 3(a)(b); 5(a)(b); 7(a)(b) (they will lie in a single plane if and only if they are linearly dependent); 9; 17; 21; 22
4.4 Coordinates and Basis 3–15 ODD; 25(a)(b)
4.5 Dimension 1; 3 (see solution); 4 (see solution); 7(a)(b)(c)(d); 9(a)(b)(c); 13; 15
4.6 Change of Basis 1(a)(b)(c)(d); 3(a)(b)(c); 5(a)(b)(c)(d)(e) (note that the "vectors" in this space are functions, not arrows); 7(a)(b)(c)(d)(e); 19 12
4.7 Row Space, Column Space, and Null Space 3(a)(b); 4(a)(b); 7(a)(b) (see Example 3 in the textbook for guidance; write the solutions in a format resembling that for this example); 8(a)(b) (again use Example 3 for guidance); 9(a)(b); (see solution); 10(a); 11(a)(b); 12(a)(b); 13(a)(b); 27 (hint: examine the interrelationships between theorem 2.3.8(a) and (b), definition 1 in section 4.3, and theorem 4.5.4)
4.8 Rank, Nullity, and the Fundamental Matrix Spaces 1(a)(b); 3(a)(b)(c); 5(a)(b)(c); 7(a)(b)(c); 9(a)(b)(c)(d)(e)(f)(g); 10; 13(a)(b); 19(a)(b)(c)(d); 30
Chapter Test 3

(Sections 4.1–4.8)
This will be a take-home exam. Chapter 4 homework will be due Thursday 1 December at the beginning of class. See the solutions.
5.1 Eigenvalues and Eigenvectors 3; 5(a)(b)(c)(d); 6(a)(b)(c)(d); 7; 9; 11; 38(a)
5.2 Diagonalization 1–13 ODD; 22; 25; 26(a)(c) 37
5.3 Complex Vector Spaces Optional (if you think you need practice with complex numbers): 1–9 ODD
Required: 11–17 ODD
29 (this is one of the quantum mechanics applications I've mentioned in class)
5.4 Differential Equations (No assignment)
5.5 Dynamical Systems and Markov Chains (No assignment)
6.1 Inner Products 1; 3; 9; 10; 11; 13; 14; 21; 22; 23; 37
6.2 Angle and Orthogonality in Inner Product Spaces 1(c); 2(c); 3; 5; 6; 9; 11; 13; 21; 22; 23; 25 29
6.3 Gram-Schmidt Process; QR-Decomposition 2(a)(b)(c)(d); 3(a)(b); 4(a)(b); 11(a)(b); 27; 28; 31; 37; 43
6.4 Best Approximation; Least Squares (No assignment)
6.5 Least Squares Fitting to Data (No assignment)
6.6 Function Approximation; Fourier Series (No assignment)
Final Examination

(Sections 5.1–5.3 and 6.1–6.3)

Recommended study-guide problems

(These are sample exam-like problems for practice purposes; do not turn in with your homework)

Bring your chapter 5/6 homework to the final for up to 20 points credit! (Not extra credit!)

Exam starts at 7:15 p.m. on Tuesday 13 December
(For students with minimal study time)

Page 343 (Supplementary Exercises): 1(a); 2; 6; 7(a)(b); 9; 11; 13; 16 (hint: see proof of Theorem 5.1.4); 17

Page 399 (Supplementary Exercises): 1(a); 2; 6; 7; 8; 9 (hint: set ${\mathbf{u}} = (1,2)$ and ${\mathbf{v}} = (3,{-1})$ and define $\left\langle {u,v} \right\rangle = {w_1}{u_1}{v_1} + {w_2}{u_2}{v_2}$, then either solve the system $\left\langle {u,v} \right\rangle = 0$, $\left\langle {u,u} \right\rangle = 1$, and $\left\langle {v,v} \right\rangle = 1$ for ${w_1}$ and ${w_2}$, or show that this is impossible); 10 (use the general inner-product definition of $\cos \theta$, not the Euclidean dot product); 12(a); one of 18 or 19 (hint: review section 7.2 of the calculus text for helpful identities, or see section 7 of this handout)
(For students with additional study time) The above plus some or all of

Page 343 (Supplementary Exercises): 3(a)(b)(c); 4; 8(a); 9; 10; 12(a)(b); 14 (hint: the degree of the characteristic polynomial is equal to $n$; what is the minimum number of $x$-intercepts on the graph of an odd-degree polynomial when you think about end behavior?); 15; 20

Page 399 (Supplementary Exercises): 1(b); 5; 11(a)(b); 12(b); 13; 18 or 19 (whichever one you haven't already done)

Additional problems taken from the unassigned homework exercises from chapters 5/6

## Course Handouts and Study Aids

• PDF (Adobe® Acrobat Reader™) is the best format to use if you want to print on paper (for example, to replace a lost copy).
• HTML files are not, for the most part, printer-friendly; this is the best format for on-screen reading, and if you can read these words, you already have the software!
• DOC and DOCX files are in the native Microsoft® Word format; if you do not have Word, use this Word Viewer from the Microsoft web site (this software can display Word files, but cannot modify them).
• PPT files are PowerPoint® presentations; if you do not have PowerPoint, use this PowerPoint Viewer, again from the Microsoft web site.

## Will You Succeed or Fail in Mathematics?

This checklist is adapted from a handout prepared by math and philosophy instructor Steve Thomassin. It will allow you to compare your approach to a mathematics course to the approaches taken by successful … and unsuccessful … students.

Overview: This table lists typical attributes of successful and unsuccessful mathematics students. It contains three columns. First row: Column headers. Second and subsequent rows: Student attributes. Details: The first row is the table header, with column headings describing the data listed in the main body of the table. The second and subsequent rows describe specific attributes that contribute to success or failure. Column one of these rows specifies whether the attribute is related to attitude, class work, homework, or getting help. Column two of these rows contains attributes of successful students. Column three of these rows contains attributes of unsuccessful students.
Attribute Type Predictor of Success Predictor of Failure
Attitude Focus on things that are under your control. Blame things that are out of your control (the text, the instructor, or "the system") for your difficulties.
Be optimistic. Believe that you can do it. Be pessimistic. Convince yourself that you will fail.
Be positive. Find ways to make math interesting and fun. Be negative. Find ways to make math dull and painful.
Be open. See the uses, power, patterns, and magic of mathematics. Be closed. Blind yourself to math's uses and its practical and esthetic value.
Be practical. Make yourself aware of the doors that passing each math class opens to you. Be impractical. Ignore the doors that open when you pass a math class.
Class Work Attend every class. Aim for perfect attendance, even if you already know it all. Be absent often. Dig a hole so deep that you cannot climb out except by dropping the course.
Be focused. Concentrate on the math topic at hand. Be mentally elsewhere. Daydream. Talk. Distract and annoy neighboring students.
Take good notes. Solve problems along with the instructor. Avoid participating in the discussion. Just watch the instructor.
Be inquisitive. Ask questions so that the instructor knows what you would like to learn more about. Be uninterested. Make the instructor guess what it is that you might be confused about.
Homework Be regular. Always do at least some homework before the next class, and finish by the due date. Be sporadic. Do homework only when it easily fits your schedule.
Invest time. Spend double to triple the amount of in-class time. Invest little time. Spend less time doing homework than you spend in class.
Review notes; read text; do all assigned problems (maybe even more), and check the answers. Ignore notes and text explanations; try a few problems, and don't bother checking to see if they are right.
Getting Help When needed, take advantage of all opportunities: study groups, tutors, instructor office hours. Even when lost, never seek assistance.

Michael Bowen's VC Course Pages: Math V22, Fall 2016