Math 13 – Linear Algebra Spring 2016

Section #2739 Meets: Tuesday/Thursday 8:00 a.m. – 9:20 a.m. Room: MC 67

Instructor: Gail Edinger Office: MC59 Campus Extension: (310) 434 – 3972

Office Hours: Monday & Wednesday 11 a.m. - 12 noon.; Tuesday 6:50 -7:50 a.m Math 13 Problem Sessions: Monday 8 – 9 a.m. Math Lab Quiet Study Room MC 84B

*Other times by appointment

Email: edinger_gail@smc.edu

Note: Due to problems with unwanted emails and student emails going to spam, please put the following in the subject line of any email: Your name- math 13 (for example Gail Edinger- Math 13). This will help me find the emails sent to spam. If you do not use this subject line I will probably not read your email.

Homepage: http://homepage.smc.edu/edinger_gail/ then follow the math 13 link.

Course Description: Topics include matrices and linear transformations, abstract vector spaces and subspaces, linear independence and bases, determinants, systems of linear equations, and eigenvalues and eigenvectors.

Prerequisites: Math 8

Required Text: Larson & Edwards, Elementary Linear Algebra, 7th

edition, Houghton Mifflin,

2012

Calculator Policy: Calculators will not be used on exams. It may be useful to have a calculator

for some of the homework problems.

Attendance: Attendance is expected and encouraged. I will take attendance at every class. In accordance with SMC policy, if you miss all or part of 4 or more classes, you may be withdrawn for non-attendance, regardless of your current grade in the class.

If you must miss a class, be sure to get the notes and any announcements from a classmate as soon as possible. You are responsible for any announcements and changes to the syllabus.

If you decide to drop this class, you are responsible for doing the paperwork, do not just stop coming. According to SMC policy students are responsible for dropping themselves, the instructor is no longer involved in the process, there is no need to ask me to drop you. Students are also responsible for knowing all dates and deadlines for the withdrawal process. Ignorance of the required dates is not an excuse for missing the deadline.

Outline: There is a course outline attached. We will try to stay as close as possible to this

schedule, but there may be changes to this schedule. At the end of each class we will verify what homework should be completed by the next class.

Homework: Homework is an important part of your success in this class. This includes not only completing the assigned problems, but carefully reading the section. There is a course outline and assignment list attached. You are expected to complete all of the assigned problems and any definitions, terms, theorems, techniques and ideas presented in the homework or reading are considered material that could be covered on an exam. This includes information that was not directly presented in class. Note that we do not have time to cover EVERY term and theorem presented in the text, so you will be responsible for reading and learning all that is there. This is a very important part of your success.

The assigned homework will be collected at the beginning of every class. From that a random selection of problems will be chosen for grading. (No, I will not tell you which ones.) Points will be assigned to total 20 points. From that 20 points, deductions may be made if the assignment is not complete. If any problem (not just the graded ones) does not include supporting work, the answer is copied from the back of the answer key, or copied from another then the ENTIRE assignment will be given a grade of 0%. Homework should be neat with pages stapled. (ONLY STAPLED) Work that is not stapled will not receive credit. Please do not turn in your entire homework notebook.

The work is due at the beginning of class on the assigned date. If you arrive for class late or try to turn in the work after we have started class (perhaps trying to finish it during class…) it will not be accepted.

There are no acceptable excuses for late work. Please don't waste my time or yours by asking for homework extensions.

If you do not turn in an assignment, if you are absent or late to class, then you will receive a grade of 0 for that assignment. At the end of the semester, the lowest 6 homework grades will be dropped, this could include assignments assigned a grade of 0 for these reasons.

All of the work should be your own. Copying work from a classmate, the internet or any answer key or any other source will be considered plagiarism. In this case all parties involved will receive a grade of 0 for the entire assignment, regardless of how much of the assignment is involved. This grade cannot be the one dropped and will count toward your final average. Please see the academic honestly policy below. You should not allow anyone to copy your work and it is probably best if you do not allow anyone to look at your final work. Anyone supplying information is considered just as guilty of academic dishonesty as someone copying information. The incident will be reported to the campus disciplinarian. Additional sanctions will be considered for anyone involved in more than one incident of academic dishonesty, including being asked to appear before the SMC disciplinary board.

Now, you may discuss the problems with other students. You should not tell them how to do the problems or ask someone to tell you. After discussing the problems all parties should separate and write the problems up themselves. If you have already solved a problem, do not let someone read yours. Think of this as like an English essay, you can discuss ideas, but all writing should be your own. If you do discuss the problems with a classmate, you must indicate with whom you discussed the work. Failure to include this information can be considered academic dishonesty.

Collected problems are a chance to get feedback on your proof writing techniques, an important part of this class.

Exams: There are 3 exams scheduled. You are expected to take the exams on the indicated date. THERE WILL BE NO MAKE-UP EXAMS GIVEN FOR ANY REASON. If you must miss

an exam, the grade on the final will be substituted for that exam. If you miss two exams, you will be given a grade of 0% for the second.

There will be a comprehensive final, scheduled for Tuesday, June 7 from 8 - 11 a.m. All students are required to take the final at this time, no exceptions except in the case of extreme and documented emergency. (Extreme emergency does not include, personal convenience, travel plans, including purchased airline tickets, other finals scheduled the same day, etc.)

Grading: The final grades will be calculated as follows:

10% Collected homework 60% 3 exams

30% cumulative final

90 – 100% = A, 80 – 89% = B, 70 – 79% = C, 60 – 69% = D, below 60% = F

All final grades will be assigned in this manner, please do not asked to be graded differently than your classmates.

Academic Honesty: The academic honesty policy of SMC will be enforced. If there is an evidence of academic dishonesty on an exam, homework assigned or in any other work, all parties involved will receive a grade of 0%, regardless of who did the original work and how much of the exam or homework assignment was involved. An academic dishonesty report will be filed with the school.

Please note that this includes, but is not limited to, use of unapproved electronic devices during exams, whether or not that device was being used for dishonest purposes.

Disabilities: Students working with the disabled student center should contact me so we can

make appropriate arrangements.

Comments:

1. This class will be approximately 70% theory and 30% computation. You will be required to memorize theorems, prove theorems you have not seen and understand proof techniques. It will be impossible to cover every theorem, lemma and definition given in the text and we will talk about the major ones in class, but you are still responsible for the entire section. Please be sure to read each section carefully.

2. You will be responsible for both learning the standard proofs given in class and in the text

as well as standard proof technique. On exams you can expect to prove statements that you have not seen before. The proofs in the homework should be done without the use of a study guide. Copying work from any source, including but not limited to study guides, the internet, books or classmates is considered plagiarism and will be dealt with under the course academic dishonesty policy.

3. Get to know each other.

4. Be sure to make good use of office hours and the math lab. In general we do not have

time to answer questions from the homework at the beginning of class. You will need to ask questions outside class.

Entry Skills: Prior to enrolling in Math 13 a student should be able to

1. Solve systems of linear equations using Gaussian elimination. 2. Write the equation of a line in parametric form. 3. Prove mathematical statements by methods including proof by contradiction and proof by

induction. 4. Evaluate, manipulate and interpret summation notation. 5. Given a function, over an interval, be able to prove algebraically the existence of its

inverse function by formally proving the function is one-to-one. 6. Be eligible for English 1. Course Objectives

1. Apply the concepts and theorems of linear algebra to show the consequences of a given definition. 2. Perform matrix computations and prove general properties of matrix algebra. 3. Express a matrix as a product of elementary matrices and an upper triangular matrix. 4. Compute the inverse, if possible, of a square matrix, and express it as a product of elementary

matrices. 5. Solve systems of linear equations using Gaussian elimination, and, where necessary, express

solutions using parameters or as a linear combination of basis vectors. 6. Apply fundamental determinant theorems. 7. Prove whether or not a set and operations form a vector space (or subspace). 8. Apply the concepts of linear independence and spanning to find a basis for a vector space. 9. Prove whether or not a function between two vector spaces is a linear transformation or

isomorphism. 10. Find the matrix representation of a linear transformation with respect to two given ordered bases. 11. Express the kernel and range of a linear transformation as a span of basis vectors. 12. Compute the eigenvalues for a matrix, find a basis for the corresponding eigenspaces, and where

possible, diagonalize the matrix. 13. Use the Gram-Schmidt process to compute an orthonormal basis of a space.

Student Learning Outcome(s):

1. Students will apply definitions and theorems of linear algebra, with topics including

linear independence, spanning, dimension, subspaces and linear transformations, to establish consequences of new definitions, prove additional results, and illustrate arguments with specific examples.

Outline

EOO= every other odd, for example 1, 5, 9, …

Date Section Assignment

2/16 1.1,1.2 1.1: 7,9,11-29 EOO, 39,41,47,49,51,53,54,61,63,67,68,77-83 odd,84

2/18 1.2 1.2: 1-17 EOO, 19-37 odd, 41,43,49,57,58

2/23 2.1 2.1: 3, 5-29 EOO, 31-47 odd, 51,57,58,59,67,74,75,76,77,85

2/25 2.2,2.3 2.2: 1-21 EOO. 23,25,29,33,35,41,47,57,60,61,65,71,73 2.3: 1,5,7-19 EOO, 41,43,47a,53,63,65,67,68,71,72,79

3/1 2.3,2.4 2.4: 1-15 odd, 19,21,22,23,27,31,35,36,49,55,57,59

3/3 2.4,3.1 3.1: 1-27 odd,31,41,43,44,49,51

3/8 3.1,3.2 3.2: 1-17 EOO, 21,25-33 odd, 37,38,47

3/10 3.2,3.3 3.3: 1,3,9,13,17-23 odd, 27,33,37,41,43,51,52,57,63,65,66,69,77,81

3/17 3.3 Continue with above and Review 4.1 on your own

3/22 4.2/Review 4.2:1-39 odd, 45 3/24 EXAM 1 Chapters 1 – 3 3/29 4.3 4.3: 1-23 odd, 29,33,36,39,43,44,45,47,51,53

3/31 4.4 4.4:1-7 odd, 9 – 25 EOO, 27, 29-41 EOO, 43, 45,47,49,51,53,57,59,60,63,64,67,69,71

4/5 4.4,4.5 4.5: 1-5 odd, 7-27 EOO, 29, 31, 33-57 EOO, 61,71,72,73,76 63,64,65,67,69,71,73,75 4/7 4.5 continue with above

4/19 4.6 4.6: 1-21 odd, 25-41 EOO, 47,49,59,67,69,71,72,76

4/21 4.7 4.7: 1,5,9,13,15,29, 31, 37, 42,43, 44

4/26 5.1, Review 5.1:1-9 odd, 11 – 23 EOO, 35,43,45,51,53

4/28 EXAM 2 Chapter 4 5/3 5.3 5.3: 1,7,13,15,17,23,27,31,39

5/5 6.1,6.2 6.1: 1,5 – 23 odd, 25,35,37,41,51,53,55,,75,79,83

5/10 6.2,6.3 6.2: 1-25 EOO,31,39,41,43,49,53,55,58,59,61,63, 67

5/12 6.3, 6.4 6.3: 1-9 odd, 11,13,27,29,31,35,37,39,43,51

5/17 6.4 6.4: 1-9 odd, 11,15,17,19,23,27,29,31,33

5/19 7.1 7.1: 1-7 odd, 13, 17-25 odd, 41,43,45,55,59,65,73,77

5/24 EXAM 3 Chapter 5, 6 5/26 7.2, 7.3 7.2: 1,5,7-17 odd, 23, 25,27,29,31,37,43,47

5/31 7.3,7.4 7.3: 1-17 odd, 25,29,35,41,47,53,55,57

6/2 Finish 7/review

6/7 FINAL 8 a.m. -11 a.m.. Comprehensive Final

SECTION 1.1 – LINEAR SYSTEMS

A system of m linear equations in n variables is a set of m equations in n unknowns: a11x1 + a12x2 + a13x3 + … + a1nxn = b1

a21x1 + a22x2 + a23x3 + … + a2nxn = b2

. . . am1x1 + am2x2 + am3x3 + … + amnxn = bm

A solution to this system is an ordered n-tuple (x1, x2, … xn) that works in all equations.

Example: The system of two equations in two unknowns 3x – 4y = -1 2x+ 5y = 7

has the solution (1, 1).

You should know how to solve linear systems in two variables, if you need to review, consult an intermediate algebra text.

What are the possible solution types for a linear system in two variables?

Consider the following examples:

x 2 y 9

1 x

2 y 3

3 3

and

x 2 y 9

1 x

2 y 7

3 3

In the case of infinite solutions, always write the solution form in parametric form. Here let x = t. (you could also say y = t, how would the solution change?)

In higher dimensions we always have the same 3 possibilities.

For a linear system in n variables, precisely one of the following is true: 1) The system has one solution. 2) The system has infinite solutions. 3) The system has no solutions.

Example: x + y + z = 5

2y – z = -7 z = 3

This form is easy to solve using back substitution. A system in this form is said to be in row- echelon form.

Example: 9x + 3y + 4z = 7 4x + 3y + 4z = 8 x + y + z = 3

We would like to reduce this to an equivalent row echelon system to solve. (Two systems are called equivalent if they have the same solution set.)

The process of reducing this to row-echelon form is called Gaussian Elimination. The operations that you can use that lead to a series of equivalent systems are:

1) Interchange 2 equations. 2) Multiply an equation by a non-zero constant. 3) Add a multiple of one equation to another.

We will solve this one together using the Gaussian elimination algorithm. Note the steps and notation carefully. Yes, at time you might find "short cuts" that will lead you to the solution, but that is not the goal here. The goal is to learn and be able to apply the algorithm. If you are asked to use the algorithm for solution of a system and you do not use the correct steps or notation, your solution is considered incorrect, even if you get the same answer at the end of the problem! So here we go...

Suppose Gaussain Elimination resulted in one of the following systems. What could you say about the solution sets?

4 x1 3x2 2 x3 1

2 x2 5x3 4

3 0

x1 2x2 x3 7

3x2 x3 0

0 0

Example:

For what values of k does the following system have one solution, infinite solutions and no solutions.

kx + 2ky + 3kz = 4k

x + y + z = 0 2x - y + z = 1

Section 1.2 – Another method of solving linear systems. A matrix is a rectangular array of numbers.

Each entry aij is identified by its row and column position in the matrix. The size of a matrix is given by the number of rows and columns. For example a 3 x 2 matrix would have 3 rows and 2 columns. The entries in the matrices we will be studying will be real numbers, but they can also be complex numbers.

If we have a linear system, we can use a matrix to represent the system.

x + y + z = 6 x -2z = -5 x + 3y = 7

This can be represented by the augmented matrix:

The system x + y + z = 0 x - 2y = 0

3y + z = 0 is called a homogeneous system. A system is homogeneous if all of the equations are set equal to zero. These are special systems as we know without even solving that there is always at least one solution. Why?

These systems can be represented by augmented matrices, but are sometimes just represented by their coefficient matrices.

We can use matrices to solve a linear system using a process called Gaussian Elimination. This process is the same as in section 1.1 and produces a series of equivalent linear systems until a solution is found. Again it is important that you follow the algorithm!

Matrices are manipulated by using the three elementary row operations. (Note the similarity to the three operations for producing equivalent systems from section 1.1)

Elementary Row Operations:

1. Interchange two rows. 2. multiply a row by a non-zero constant 3. Add a multiple of one row to another.

Two matrices are called row equivalent if one can be obtained from the other by a sequence of

elementary row operations.

It is very important that you know the above operations and limit your matrix work to these.

Go back to the two examples above. Due to space considerations, we will do these on a separate sheet of paper.

Row- Echelon Form

A matrix is in row-echelon form if:

1. All of the rows consisting entirely of zeros, if any, are at the bottom.

2. For each row that does not consist entirely of zeros, the first non-zero entry is 1 (this is called the leading 1 or the pivot.)

3. For 2 successive non-zero rows, the leading 1 in the higher row is farther left than the leading 1 in the lower row.

Examples:

Reduced Row Echelon Form:

Row echelon form plus all entries in a column of a leading 1 are zero.

Examples:

A system of equations can be solved using Gauss-Jordan elimination.

This takes the matrix to reduced row echelon form.