LINEAR ALGEBRA SYLLABUS Fall 2019 Professor Sormani
MAT313 Elements of Linear Algebra: 4 hours, 4 credits. Vector spaces, systems of linear equations, determinants, linear transformations, and matrices. PREREQ: MAT 176. With Departmental permission, MAT 176 may be taken as a COREQ.
Meeting Times: 6:00-7:40 pm Mon/Wed in Gillet 305
Course Webpage:
https://sites.google.com/site/professorsormani/teaching/linalg-f19
Google "Professor Sormani" then select "Teaching" then select "Linear Algebra"
Instructor : Professor Sormani Email: sormanic at@ gmail.com
Webpage: https://sites.google.com/site/professorsormani/
Office: Gillet 200A Office Hours: 7:40-8:20 pm Mon/Wed
Grading Policy
Expectations: Students are expected to learn both the mathematics covered in class and the mathematics in the textbook.
Completing homework is part of the learning experience. Learning to learn from a book is a crucial life skill.
Homework: Approximately four hours of homework will be assigned in each lesson and due on the first meeting of each week.
The solutions to the homework are provided on the textbook website so check your work before submitting it. Come to office
hours before class if you are unsure. Homework assignments will be listed on the course webpage (below).
Quizzes: Quizzes will be randomly given in any lesson at any time based upon problems similar to the homework due that day or earlier in the week. This includes problems which involve proofs. Up to three quizzes can be retaken during office hours within two weeks for a max score of 80%.
Midterm Exam: A midterm can be retaken for a max score of 80% within two weeks.
Final: A missed final is an incomplete in the course only if one already has a passing average.
Grading: Quizzes (30%), Midterm Exam (30%), Final Exam (40%)
Materials, Resources and Accommodating Disabilities:
Textbooks:
A First Course in Linear Algebra, an interactive online textbook by Beezer
Linear Algebra, a free textbook by Jim Hefferon with answers to exercises
Tutoring: Julinda Mujo in the Math Lab can help with this course
Accommodating Disabilities: Lehman College is committed to providing access to all programs and curricula to all students. Students with disabilities who may need classroom accommodations are encouraged to register with the Office of Student Disability Services. For more info, please contact the Office of Student Disability Services, Shuster Hall, Room 238, phone number, 718-960-8441
Accommodating Holidays: If you have a holiday during a lesson or extra lesson, let me know, and something will be arranged.
Names/Gender: We will use last surnames in this class. You may call me Sormani or Professor.
Respect: All students will treat each other with respect and dignity. Let me know if you have concerns.
Course Outcomes
1. Learn to prove theorems. Students should be able to observe connections between different topics in the course description above, especially systems of equations, matrices and vector spaces.
2. Learn to perform calculations. Students must learn how to perform calculations in different settings. They should be comfortable doing algebra on matrices, solving systems of linear equations and working with vectors.
Math Major Outcomes incorporated into MAT313
A. Perform numeric and symbolic computations
B. Construct and apply symbolic and graphical representations of functions
C. Model real-life problems mathematically
E. State and apply mathematical definitions and theorems
F. Prove fundamental theorems
Department of Mathematics and Computer Science, Lehman College, City University of New York
Please email me your name, major, career plans, a photo, and any concerns or holidays.
Schedule:
The schedule will be available on the course webpage listing each lesson’s topics with the homework to be completed after the lesson beneath the lesson. Extra lessons are inserted in case there are class cancellations due to weather or illness later in the semester. There will be 28 Lessons worth of material. When possible the lessons will link to sections of the online interactive text. If you miss class then you can learn directly from the textbooks.
(Wed 8/28) Lesson 1: Vectors in Euclidean Space and Set Notation
HW: Read Hefferon Ch One Part II.1 Do 1.1-1.6, 1.9 Read Part II.2 Do 2.11, 2.12, 2.14, 2.15 The solutions are in the solutions manual. Note that if you go to the contents of Hefferon and click on Chapter 1 Part II it takes you right to the correct place.
Also Read Beezer Preliminaries on Set Notation
No Lesson: (Mon 9/2) Holiday
(Wed 9/4) Lesson 2: Solving Linear Systems (Quiz 1 on Lines postponed)
HW: Read Beezer: What is Linear Algebra and then Read Hefferon Ch One Part I.1 and do exercises: One.I.1/ 1.17, 1.18, 1.29, 1.30, 1.35,
(Th 9/5) Lesson 3: Complex Numbers
HW: Read Beezer Preliminaries on Complex numbers and then practice at IXL
(Mon 9/9) Lesson 4: Solving Linear Systems with Matrices (Quiz 1 on Lines, Quiz 2 on Solving Linear Systems)
HW: Read Hefferon Ch One Part I.2 Do exercises: One.I.2/ 2.17, 2.18, 2.19, 2.21, 2.22, 2.25, 2.26, 2.27
also Read Beezer: Solving Systems of Linear Equations
(Wed 9/11) Lesson 5: Reduced Row Echelon Form and Homogeneous Linear Systems
HW: Read Beezer: Reduced Row Echelon Form and do exercises C10-C33
You might also consult the Hefferon Ch One III which has full solutions in more detail
(Mon 9/16) Lesson 6: A Matrix times a Vector and Null Spaces (Quiz 3 on Reduced Row Echelon Form)
HW: Read Beezer: Homogeneous Systems and do C21-C31 but go one step further factoring out free variables.
or Read Hefferon Ch One Part I.3 Do exercises: One.I.3/ 3.14, 3.15, 3.16, 3.18, 3.20, 3.23
(Wed 9/18) Lesson 7: Identity Matrices, Permutation Matrices, Nonsingular Matrices,
HW Read Beezer Nonsingular Matrices and do C50, C31, C32, C33
Read the beginning of Beezer Eigenvalues and Eigenvectors and do Example SEE carefully.
Practice multiplying all the matrices times vectors that were done in class
Email me if you did not receive the class photos
Do not read Beezer Vector Operations but do C10,C11,C12, C13,
(Mon 9/23) Lesson 8: Linear Combinations, Spans, Linear Independence, (Quiz 4 on Null Spaces and Matrix times Vector)
HW Read Beezer Column Spaces do C31, C35, M10, M20
Read Beezer Linear Independence do C20-25
(Wed 9/25) Lesson 9: Finding a Basis (using Pivot columns) and an orthonormal basis (using Gram Schmidt)
Do not read Beezer Vector Operations but do C12,C13,C14, C15
Read Beezer Linear Independence do C31, C32, C33
(Mon 9/30) Extra Lesson 10: Proofs with Vectors (Quiz 5 on Linear Independence)
Prove that if A is a matrix and v,w are vectors and Av=0 and Aw=0 then A(v+w)=0 and A(rv)=0.
Prove that a null space is closed under vector addition (as in class) and scalar multiplication (for extra credit due 10/14)
Prove that a span is closed under scalar multiplication (as in class) and vector addition (for extra credit due 10/14)
Prove that if B is a matrix then B(v+w)=Bv+Bw and B(rv)=rBv
Read Beezer Vector Operations but do T13,T17,T30
(Wed 10/2) Lesson 11: Dimension, Nullity, Range, and Rank of a Matrix and Review for Midterm
The range of a matrix is the column space which is the span of its columns.
The dimension of a span of vectors is the number of vectors in its basis.
The rank of a matrix is the dimension of the range (so it is the number of leaders or pivot columns)
The null space of a matrix A are the vectors v such that Av=0
To find the null space solve the homogeneous system. You will see the null space is a span of direction vectors.
The nullity is the dimension of the null space so it is the number of free variables (because the direction vectors of free variables are always linearly independent).
(Mon 10/7) Lesson 12: Midterm Exam on material in Quiz 1-5 plus Extra Credit on Lessons 10-11.
The midterm exam has five parts, each part is like one of the quizzes. To avoid doing too much work, students may refer to early parts in order to complete each part. A row reduction completed for one part might be used for another part if the numbers match.
(Wed 10/9) No Lesson: Lehman Holiday
(Mon 10/14) Extra Lesson 13: (Extra Credit Quiz on Vector Proofs) Dr. Murillo speaks
Catch up on retakes of Quiz 1-5 if needed before class starting at 5pm. There are most 3 retakes all semester. Each retake was meant to be for an emergency or feeling unwell during a quiz, or a conference or other personal reason, there is no reason to tell me why. Three such occasions is more than anyone should need. Some students already used up their three retakes in the first five quizzes and now have no space for future emergencies.
(Wed 10/16) Lesson 14: Matrix Addition and Multiplication
Practice Matrix Multiplication here,
Prove (A+B)v=Av+Bv when A and B are 2x3 matrices and v is a 3 vector
Prove (A+B)C=AC+BC when A B and C are 2x2 matrices
Find a 3x3 matrix which switches rows 1 and 2: r1 <-> r2
(hint just do this row action to the identity matrix)
Find a 3x3 matrix which adds twice row 2 to row 1: r1->r1+2r2
Find a 3x3 matrix that multiples row 3 by 5: r3 -> 5r3
(Mon 10/21) Lesson 15: Inverses of Matrices (Quiz 6 on Matrix Mult)
Read Hefferon Ch Three IV.4/ Do 4.12, 4.13, 4.14, 4.20, 4.30, 4.32, 4.33,
find the inverse of a 2x2 matrix in general
(Wed 10/23) Lesson 16: Inverses of Matrices (Make up for Exam I, EC on Inverses)
Extra Credit on Inverses if not yet done (see class photos)
(Mon 10/28) Lesson 17: Determinants (Quiz 7 on finding Inverses)
HW: Hefferon Read all of Ch Four I.2 Do 2.8, 2.9, 2.10, 2.13,
(Wed 10/30) Lesson 18: Properties of Determinants
Read all of Hefferon Ch Four I.3 Do 3.17, 3.18, 3.30, 3.33,
(Mon 11/4) Lesson 19: Inverse, Determinant, Trace, Transpose, and Adjoint of a Matrix (makeups in class)
Extra Credit on these topics if not yet done in class (see class photos)
(Wed 11/6) Lesson 20: Eigenvalues and the Characteristic Polynomial (Quiz 8 on finding Determinants by row reduction and other methods)
HW to submit as part of Quiz 9 on Monday: Read Hefferon Chapter Five II.3 Examples 3.7 and 3.8, Then find the eigenvalues of the matrices in 3.22, 3.23, 3.24, 3.25, 3.26
and check the answers in the solutions manual.
(Mon 11/11) Lesson 21: Eigenvalues, Eigenvectors and Eigenspaces (Quiz 9 on Eigenvalues)
Read Hefferon Chapter Five II.3 Examples 3.7 and 3.8, Then find the eigenvectors of the matrices in 3.22, 3.23, 3.24, 3.25, 3.26
and check the answers in the solutions manual.
(Wed 11/13) Lesson 22: Similar Matrices and Diagonalization
Read Hefferon Ch Five II.1 Do 1.4, 1.10, 1.11,1.12, Read Ch Five II.2 focusing on Defn 2.1, Ex 2.2, Ex 2.3 Read Ch Five II.3 and practice finding eigenvalues and eigenvectors: 3.20, 3.21, 3.22, 3.23, 3.24, 3.28, 3.31, 3.35, 3.41, see the solutions manual for help.
(Mon 11/18) Lesson 23: Power Method and Symmetric Matrices (Quiz 10 on Eigenvectors)
For the power method see http://college.cengage.com/mathematics/larson/elementary_linear/5e/students/ch08-10/chap_10_3.pdf
Do 11, 13, 15, 17, 19 at the end of Chapter 10.3
(Wed 11/20) Lesson 24: Vector Spaces and Subspaces
Read Hefferon Chapter Two I.1 and do 1.17, 1.18,1.20b,1.21b,1.22abcd
Hand in this homework to count towards Quiz 11 on Vector Spaces
(Mon 11/25) Lesson 25: Proofs with Vector Spaces and the Basis of a Vector Space (Quiz 11 on Vector Spaces)
Read Hefferon Chapter Two II.1 III.1 and III.2, no assigned problems (everything was on the quiz)
(Wed 11/27) Lesson 26: Linear Maps (Homomorphisms) and Transformations
Read Hefferon Chapter Three II.1 (read all of it) Do 1.18abc, 1.19abd, 1.20ab, 1.25
(Mon 12/2) Lesson 27: Applications of Linear Maps and Transformations
Read Hefferon Chapter Three II.1 (read all of it) Do 1.27abc, 1.27abcd, 1.40ab
Read Hefferon Chapter Five II.3 (read only Defn 3.1 Examples 3.2-3.4) Do 3.30, 3.39, and 3.42
(Wed 12/4) Lesson 28: No meeting (students may work together to complete HW from last two lessons together which will be handed in as part of Quiz 12.
(Mon 12/9) Lesson 29: Infinite Dimensional Spaces, Analog to Digital (Quiz 12 on Linear Transformations)
Retakes should be completed today after class.
(Wed 12/11) Lesson 30: Review
The final will be given during finals week based on Quizzes 6 - 12.
Final is on Monday Dec 16 6:15-8:15 pm