# Elements of Linear Algebra

 Elements of Linear Algebra, MAT313, 4 credits Section ZI81 7:45-9:25 T Th  Prerequisite: MAT 176  Text: Kolman, Introductory Linear Algebra with Applications, 6th edition, Prentice Hall  Course Webpage: select Elements of Linear Algebra from my webpage. Professor C. Sormani Office Hours: 5:30-6:00pm TuTh                         9:30-10:00pm TuTh  Office: Gillet Hall 200B  Email: sormani@comet.lehman.cuny.edu  Phone: 718-960-7422  Webpage: http://comet.lehman.cuny.edu/sormani

Grading Policy: Exam I: 25%... Exam II: 25%... Project: 20%... Classwork: 5%... Final: 25%...

Each lesson has a corresponding reading assignment and homework problems. The problems will not be collected.  However, it is expected that you will do the homework and check your answers in the text.  You should ask questions in class and in my office hours when you have difficulty.  Problems with a * will be put on the board by students and gone over at the beginning of class. .

There will be two exams given in class. The best way to prepare for the exams is to review your class notes, the corresponding chapters in the textbook and practice problems without consulting your notes.  Always start studying at least 4 days before the exam and bring questions to the review class.  The final exam will be similar to the exams, emphasizing the material learned after the second exam but covering the rest of the semester as well. Be sure to review your old exams before taking the final. The project will be a semester long assignment with four components and due dates. The classwork will consist of in class participation and  homework problems which are written on the board before class begins.  Extra credit assignments will be given from time to time.

Syllabus: Vector spaces, systems of linear equations, determinants, linear transformations, and matrices.

• Short Week 1: 1.1 Systems of Linear Equations

• 8/31: Read 1.1/ex 1, 2, 7 and pp 28-30. Do 1.1/*1,3,7,*9,*15,19,*23,*T.4 1.3/*T.11,*T13ab
• Week 2: 1.2-1.3 Matrices

• 9/5: Read 1.2, do 1.1/5, 11; 1.2/1, 5abe*,7abc*, *T1, *T4, *T5a
9/7: Read 1.3 pages 18-20, ex 9, linear systems, and ex 13,
Do 1.2/5c, T5bc, *T6ab, 1.3/1a, *5ab, *9, *15, 19, *27, T1, *T4, T7,
• Week 3: 1.4 Properties of Matrices

• 9/12: Read theorems and defn in 1.4.  Do 1.3/1b, 5c, 29, *T6 1.4/*2, 5, *15, T1,*T2,
9/14:  read 1.4: prove Thm 1.2 a using summation notation, Example 9,  Do: 1.3/T2, *T3a, *T10, T13c, 1.4/*7, 9, *13a, *19, *T3, *T14
Read the intro to each of the projects listed below and choose one before 9/19. You should read all the intros and then go a little further in your chosen topic.
• Week 4: 1.5-1.6 Solutions and Inverses

• 9/19: Read 1.5, Do 1.3/*T9a, T3b, 1.4/*T19, 1.5/1,3,7*,9a*,9b*, T9*
9/21: Read 1.6, 1.3/ T13 (with n=3 and m=4), 1.4/18, *T32, 1.5/*9c, 15a, 17a, 1.6/1, *3, *5b, *7a, 7b, *19, *25
• Week 5: 2.1-2.2 Determinants

• 9/26: Read 2.1, Do 1.6/5ac, *20, 2.1/1,5,*8,*15a, *T3
9/28: Read 2.2 up to page 112, Do 1.3/27*, 1.4/*T21, T23, *T26, 1.6/7c, 2.1/3, 19a, *T13 2.2/*3a,*9
• Week 6: 2.3, Review and Exam I on Thursday.
There will be five problems: 1) solving a linear system, 2) matrix multiplication with an application (like 1.3/example 9) 3) a proof similar to the ones in 1.4 (know the theorems and defns), 4) finding an inverse, 5) taking determinants and using this information

• 10/3: Review Homework: 1.3/28,29,30 1.5/T15, T16a, 1.6/T9, 1.6/9bc, 1.5/9abc, 2.2/3b, 11c
10/5: Exam I: On Chapters 1-2
Phase 1 of the project is due on 10/12.
• Short Week 7: 3.1-3.2: Vectors

• 10/10: no class monday schedule
10/12: Read: 127-130, 132-139, 141-149. Do: 3.1/2a, *5a(sketch), 9a, *19a, *21a, 3.2/1a, 7a*, 11a, 15*, 21a, 27a, T7
Phase 2 of the project is due on 10/17.
• Week 8: 3.3-3.4: Linear Transformations and Computer Graphics

• 10/17: Read 3.3, Do: 3.1/5c(sketch), 9b, 19b, 21d 3.2/T6, 3.3/*T2, *1ac, *17, *19, *23
10/19: Read 3.4 and 3.6 pages 187-191, Do: 1.6/9a, 3.2/7b, 23, 27b, *T8, 3.3/*3abc, 25 3.4/*1c, *3, *4,
Extra Credit: Find the standard matrix for a transformation that rotates a 3 vector around the z axis by 90 degrees and call it A, find the standard matrix for a 3 vector that rotates a 3 vector around the x axis by 90 degrees and call it B, Is AB=BA? Check your answer with a picture and by taking the products.
• Week 9: 4.1, 4.2, 6.1: Vector Spaces

• 10/24: Read 4.1, Do 3.4/1d, 4.1/*2, *3, 5, 8, *9, *11, 17, *T1, *T4
10/26: Read 4.2 and 6.1, Do: 4.2/ 1, *7, *15, *16, *19, *21a, *T3, 6.1/ 1, 3a, *9, 13
• Week 10: 4.3-4.4 Basis and Dimension

• 10/31: Read 4.3 up to Example 12, Do: 4.2/*20, *T5, 6.1/ *T3, 4.3/*1, *5, *T1, T4, *T7
11/2: Read 4.4, Do: 1.6/ 7c, 4.2/ *T9, 6.1/ *T5, 4.3/3, 7, *T13, 4.4/1ac, 3a, *7ab, *11, *17a,
Practice old homework for Exam II.
• Week 11: Review and Exam II on Thursday.
There will be five problems. 1) Adding vectors, finding the lengths of vectors and angles between them, sketching. 2) Graphics, sketching the images of vectors under linear transformations, finding the standard matrix. 3) Verifying that a transformation is a linear transformation. 4) Verifying that a subspace of a vector space we've studied is a subspace (checking closure). 5) Verifying that a set of vectors is linearly independant, that they span a given set, and that they are a basis.

• 11/7: Vote even if you will be late for class!
11/9: Exam II focusing on 3.1-3.3, 6.1 and 4.1-4.4, will end promptly at 9:25 pm.
Phase 3 of the project is due on 11/16.
• Week 12: 4.5-4.8 Rank and Orthogonality

• 11/14: Read 4.5 (skip Ex. 2 and 3) and 4.6, Do 4.4/17b, 4.5/1, *3, 5, 7, *9, 4.6/1, *3, *9, *11
11/16: Read 4.8, Do 1.6/*5b, 4.3/T9, 4.4/31, 4.6/13, 15, 4.8/1a*, 3*, 5*, 19*
• Short Week 13: 4.9 Orthogonal Matrices

• 11/21: Read 4.9 and 3.3 Exercise 1, Do 4.6/*17, *19, 4.8/ 7, *11, *T5, 4.9/ *2(graph), *5(row and null spaces only), *7abc, T5, Extra Credit: 4.9/T6.
11/23: no class, Thanksgiving
• Week 14: 6.2 Kernel and Range

• 11/28: Review 6.1 and read 6.2, 6.1/3a*, 9*, T4*, 6.2/1, 3*, 7*, 11*, 19, T3*, T4*, T5*,
11/30: Read 5.1 pages 291-299 and Read 4.5 Exercises 2 and 3, 1.6/7c, 4.4/ 29, 6.2/9, 5.1/ 1, 5*, 7*, 9*, 11*, T4*(3x3 case), T6*
Phase 4 of the project is due 12/7.
• Week 15: 5.1-5.2 Eigenvalues and Eigenvectors

• 12/5: Read 5.1 pages 299-309, 5.1/13*, 15*, 17*, 23*,
12/7: Read 5.2, and pages 356-358, 5.1/19*, 21*, 25*, T12*, T15*, 5.2/1, 5*, 7*,9*, T1, T5*
Extra Credits due on Tuesday Dec 19:
• I. Read 8.6, Do 8.6/1, 2*, 3, 4*, T1*
• II. Read 8.8, Do 8.8/1, 2*, 5, 6*, 23, 24*
• Short Week 16: Catch up and Review

• 12/12: Review for the final: You may bring one 8.5x11in sheet of paper filled with notes on one side to the final. Please bring a xeroxed copy of the sheet to hand in with the exam. Study the following topics and create one page of notes for each, then compile these notes into a single sheet.
• Solving Homogeneous systems, Reduced Row Echelon, finding independant and dependant variables, writing a soln as a linear combination of vectors, relationship to nonhomogeneous systems, Read 4.5, Example 1 and see page 242, This will be tested as part of an eigenvalue problem, kernal problem and/or null space problem. Study this first!
• Using and Finding Inverses of Matrices to solve linear systems of equations: Read 1.6, 5.2, Do 1.6/25, p87/17, p86/13, 7
• Linear Transformations, Kernals and Ranges, Standard Matrices, Null Spaces and Column Spaces: Read 4.6, 6.1 and 6.2, Do 6.2/Examples 11 and 12.
• Vector Spaces, Basis, span, linear independence, orthonormal basis, Gram Schmidt: Read 4.3, 4.4, 4.8, Do 4.4/Examples 2 and 4, use procedure on p273, Do 4.8/ Examples 4 and 5.
• Eigenvalues, Eigenvectors, Diagonalization, using determinants and solving homogeneous systems to find eigenspaces: Read 2.1, 4.5, 5.1-5.2, Do 2.1/ Examples 5, 6 and 7, 4.5/Example 2, 5.1/Example 6, procedure page 305, 5.1/1, 7, 13, 5.2/Example 5.
• Proofs and Theory: Summation Notation, Defn of a Linear Transformation, Defn Vector Space, Defn of Matrix Multiplication with sum notation, Read p28-29, 1.3, 1.4, 4.1, 6.1, Do 1.3/T4, 1.4/T7, 4.1/Examples 3 and 9, 5.2/T1, 6.1/T4.
• Extra Office hour 8-9pm Tuesday Dec 19

• 12/19: Bring questions.
• Final: Thursday Dec 21 8-10pm G205, You may bring one 8.5x11in sheet of paper filled with notes on one side to the final. Please bring a xeroxed copy of the sheet to hand in with the exam. No Calculators. There will be five multipart problems. See review topics above.
Projects:
• Marcov Chains (only if you have taken probability)

• Intro: read the first paragraph in 8.3
Phase 1: read pp 439-442 and review probability, do 8.3/1-4
Phase 2: finish reading 8.3, do all odd problems, find further reading
Phase 4: do problems from the further reading
• LU- Factorization (Comp Sci Majors)

• Intro: read the first paragraph in 9.3
Phase 1: read 9.2 and do exercises 9.2/1,3,5
Phase 2: do all odd exercises in 9.2
Phase 3: read 9.3 and do exercises 1, 3, 5
Phase 4: complete all odd exercises in 9.3
• Linear Economic Models

• Intro: read the first page in 8.5 and Example 1
Phase 1: read 461-464, do exercises 1, 3 and 5
Phase 2: finish 8.5, get further reading from a library and do all odd exercises
Phase 4: do problems from further reading
• Linear Programming

• Intro: read the first page in 7.1 and Example 1
Phase 1: read 7.1 pages 371-379 and do exercises 1, 3, 11, 13
Phase 2: finish reading 7.1 and do all odd exercises
Phase 3: read 7.2 pages 389-395 and do exercises 1,3
Phase 4: finish 7.2 and complete all odd exercises (in future: 7.2/5, 7, 15
• Graph Theory

• Intro: read the first page in 8.1 and Example 1
Phase 1: read 8.1 pages 417-422 and do exercises 1, 3, 5
Phase 2: finish 8.1 and do all odd problems
Phase 3: do problems 8.1/ 2,4,6,8, T.2, T.3
Phase 4: complete all even problems