**Course: MATH 115A, Linear Algebra**, Lecture 6, Spring 2015

**Prerequisite:** MATH 33A, Linear algebra and applications.

**Course Content:** Linear independence, bases, orthogonality, the Gram-Schmidt process,
linear transformations, eigenvalues and eigenvectors, and diagonalization of matrices. This course
should develop your ability to write rigorous proofs.

*Last update:* 20 May 2015

**Lecture Meeting Time/Location:** Monday, Wednesday and Friday,
2PM-250PM, MS 5127

**Instructor:** Steven Heilman, heilman(@-symbol)ucla.edu

**Office Hours:** Mondays, 9AM-11AM, Wednesdays 1PM-2PM, MS
7370

**TA:** Sangjin Lee, sangjinlee(@-symbol)math.ucla.edu

**TA Office Hours:** Tuesdays 3PM-4PM, Wednesdays 4PM-5PM, MS 3919

**Discussion Session Meeting Time/Location:** Tuesdays and Thursdays,
2PM-250PM, MS 5137

**Required Textbook:** __Linear Algebra__, Friedberg, Insel and Spence, 4th
Ed., **Custom Edition for UCLA**

**Other Textbooks (not required):** __Linear Algebra: an introductory approach__, C. W. Curtis

**TA Course Website:** here

**First Midterm:** April 24, 2PM-250PM, KNSY PV 1200B

**Second Midterm:** May 18, 2PM-250PM, KNSY PV 1200B

**Final Exam:** June 11, 1130AM-230PM, PAB 1434A

**Other Resources:**
115A, Tao, Fall 2002: I would highly recommend
reading these lecture notes. My own lecture notes below are meant to be a more condensed presentation of similar material. So, if you
prefer a more thorough treatment, I recommend these notes (and the book).

An
introduction to mathematical
arguments, Michael Hutchings,
An Introduction to Proofs,
How to Write Mathematical Arguments

**Exam Procedures:** Students must bring their UCLA ID cards to the
midterms and to the final exam. Phones must be turned off. Cheating on
an exam results in a score of zero on that exam. Exams can be
regraded at most 15 days after the date of the exam.

**Exam Resources:** Here
is a page containing old exams
for a similar linear algebra course. Occasionally these exams will cover
slightly different material than this class, or the material will be in a
slightly different order, but generally, the concepts
should be close if not identical.

Here are solutions to
this
second midterm. (Note this practice midterm is much longer than our
exam.)

Here are solutions to
this
practice final. (Skip question 7; also questions 5,6 and 8 are a bit
challenging.)

**Homework Policy:**

- Late homework is not accepted.
- The lowest homework grade will be dropped. This policy is meant to account for illnesses, emergencies, etc.
- Do not submit homework via email.
- There will be 8 homework assignments, assigned weekly (on Thursday)
and turned
in at the
**beginning**of each discussion session on the following Thursday. - A random subset of the homework problems will be graded each week. However, it is strongly recommended that you try to complete the entire homework assignment.
- You may use whatever resources you want to do the homework, including computers, textbooks, friends, the TA, etc. However, I would discourage any over-reliance on search technology such as Google, since its overuse could degrade your learning experience. By the end of the quarter, you should be able to do the entire homework on your own, without any external help.
- All homework assignments must be
**written by you**, i.e. you cannot copy someone else's solution verbatim. I would encourage you to understand carefully how the homework solutions work, and how you would find such a solution on your own. Overusing collaborations or search technology should result in poor performance on the exams. - Label your homework with the lecture number and the discussion section number.
- Homework solutions will be written by the TA and posted soon after the homework is due.

- The final grade is weighted as the larger of the following two schemes. Scheme 1: homework (15%), the first midterm (20%), the second midterm (25%), and the final (40%). Scheme 2: homework (15%), largest midterm grade (35%), final (50%) The final grade will be curved. However, anyone who exceeds my expectations in the class by showing A-level performance on the exams and homeworks will receive an A for the class.
- We will use the MyUCLA gradebook.
- If you cannot attend one of the exams, you must notify me within the first two weeks of the start of the quarter. Later requests for rescheduling will most likely be denied.
- You must attend the final exam to pass the course.

** Tentative Schedule**: (This schedule may change slightly during the
course.)

Week | Monday | Tuesday | Wednesday | Thursday | Friday |

1 | Mar 30: 1.2, Vector spaces | Apr 1: 1.3, Subspaces | Apr 2: No homework due | Apr 3: 1.4, 1.5, Linear systems, Linear independence | |

2 | Apr 6: 1.5, 1.6, Linear independence, bases | Apr 8: 1.6, Dimension | Apr 9: Homework 1 due | Apr 10: 2.1, Linear transformations | |

3 | Apr 13: 2.1, Linear transformations | Apr 14: 1.6, Lagrange interpolation | Apr 15: 2.1, 2.2, Null spaces, range, coordinate bases | Apr 16: Homework 2 due | Apr 17: 2.2, Matrix representation |

4 | Apr 20: 2.3, Matrix Multiplication | Apr 22: 2.4, Invertibility | Apr 23: Homework 3 due | Apr 24: Midterm #1 | |

5 | Apr 27: 2.4, Isomorphism | Apr 29: 2.4, 2.5, Change of coordinates | Apr 30: Homework 4 due | May 1: 3.1-4.3, Row operations | |

6 | May 4: 3.1-4.3, Rank of matrices | May 6: 4.4, Review of determinants | May 7: Homework 5 due | May 8: 5.1, Diagonal matrices | |

7 | May 11: 5.1, Eigenvalues and eigenvectors | May 13: 5.2, Diagonalization | May 14: Homework 6 due | May 15: 5.2, Characteristic polynomials | |

8 | May 18: Midterm #2 | May 20: 6.1, Inner products | May 21: No homework due. | May 22: 6.1, 6.2, Norms, orthogonal bases | |

9 | May 25: No class | May 27: 6.2, Gram-Schmidt orthogonalization, complements | May 28: Homework 7 due | May 29: 6.3, Adjoints | |

10 | Jun 1: 6.4, Normal operators | Jun 3: 6.4, Self-adjoint operators | Jun 4: Homework 8 due | Jun 5: Review of course |

**Advice on succeeding in a math class:**

- Review the relevant course material
**before**you come to lecture. Consider reviewing course material a week or two before the semester starts. - When reading mathematics, use a pencil and paper to sketch the calculations that are performed by the author.
- Come to class with questions, so you can get more out of the
lecture. Also, finish your homework at
least
**two days**before it is due, to alleviate deadline stress. - Write a rough draft and a separate final draft for your homework. This procedure will help you catch mistakes. Also, consider typesetting your homework. Here is a template .tex file if you want to get started typesetting.
- If you are having difficulty with the material or a particular homework problem, review Polya's Problem Solving Strategies, and come to office hours.

- 1: Introduction, Fields, Vector Spaces, Subspaces, Bases, Dimension
- 2: Linear transformations, matrices, invertibility, coordinates
- 3: Matrices, row operations, determinant
- 4: Eigenvalues, eigenvectors, diagonalization
- 5: Inner Products, Adjoints, Spectral Theorems, Self-Adjoint Operators
- Linear Algebra, 115A, Spring 2015 (full set of notes)