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 inclass 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
 ISBN13: 9781118473504 (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
 ISBN13: 9781118677308 (less expensive UNBOUND (loose pages) version; binderready)

 Author: H. Anton
 Title: Elementary Linear Algebra Applications Version, Eleventh Edition
 ISBN13: 9781118434413 (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
 ISBN13: 9781118474228 (less expensive UNBOUND (loose pages) version; binderready)
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 ereader) 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
 ISBN13: 9781118464427
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 SCI223
 Monday 12 December 2016: 10:15 a.m. to 12:15 p.m. in SCI352 (may move to MCE227 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 SCI354, 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.
§  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 opensource 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 InputOutput Models  (No required assignments; extra credit only)  4 
Chapter Test 1 (Chapter 1) Recommended studyguide problems (These are studyguide 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 2Space, 3Space, and nSpace  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 pointnormal 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 studyguide problems (These are sample examlike 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; crosscheck 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; crosscheck 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 takehome 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  GramSchmidt Process; QRDecomposition  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 studyguide problems (These are sample examlike 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 innerproduct 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 odddegree 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
The documents listed below are available for viewing or download. The list below provides links to download free software to read the file formats of the various documents.
 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, printerfriendly; this is the best format for onscreen 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.
Course Handouts

Course Information (HTML)
Course Information (PDF) 
Course Requirements and Grading, Side 1 (HTML)
Course Requirements and Grading, Side 1 (PDF) 
Course Requirements and Grading, Side 2 (HTML)
Course Requirements and Grading, Side 2 (PDF) 
Tips for Success (HTML)
Tips for Success (PDF) 
Standards of Student Conduct and Classroom Rules (HTML)
Standards of Student Conduct and Classroom Rules (PDF) 
Syllabus Worksheet (DOC)
Syllabus Worksheet (PDF)  Instructor's Schedule (PDF; not really a handout; this is a copy of the printed schedule that appears on the instructor's office door)
Study Aids

Multiplication Tables (DOC)
Multiplication Tables (PDF) 
Divisibility Rules (DOC)
Divisibility Rules (PDF)  Sieve of Eratosthenes (PDF) with directions (HTML) (finds prime numbers)
 Powers of Ten Tutorial (HTML) (offsite; requires Java™ Runtime Environment [free download] to be installed and enabled on your computer)

Translating English Phrases into Algebraic Expressions (DOCX)
Translating English Phrases into Algebraic Expressions (PDF)  Multilingual Vocabulary for Mathematics (possibly helpful for students whose first language is not English) (HTML)

Basic Algebra Review (DOC)
Basic Algebra Review (PDF) 
Basic Geometry Review (PPT)
Basic Geometry Review (PDF)  Rectangular Graph Paper:

Quadratic Functions: Questions and Answers (DOC)
Quadratic Functions: Questions and Answers (PDF)  Transformations of Functions (HTML) (may require downloading and installation of free software to view all portions; see the page itself for details)

Essential Trigonometric Identities for Physics & Calculus (DOC)
Essential Trigonometric Identities for Physics & Calculus (PDF)  Polar Graph Paper:
 Math V21B Maple Scripts for 3D Visualization and Plotting Polar Functions (HTML)
 Series Convergence/Divergence Flow Chart (PDF)
 Math V22 Maple Scripts for Manipulating Matrices and Vectors (Page 1) (HTML)
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.
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 inclass 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. 