# Math 113: Linear Algebra and Matrix Theory, Spring 2013

## Tuesday, Thursday 12:50-2:05pm in 380-380W

### Teaching Staff

 Instructor Sheel Ganatra Course Assistant Graham White Office 380-382F Office 380-380R e-mail ganatra (at) math (dot) stanford (dot) edu e-mail grwhite (at) math (dot) stanford (dot) edu Office Hours Tuesday, Thurdsay, 11.15am-12.45pm Office Hours Monday, Wednesday 4-5.30pm

### Course Description and Prerequisites

Math 113, a linear algebra course, will initiate the study of vector spaces and linear maps between vector spaces. The first and most familiar example of a vector space is the set of n-tuples of real or complex numbers. However, we shall see that vector spaces and the techniques of linear algebra can be found on many sets with good notions of addition and multiplication by scalars (for example: polynomials with real or complex coefficients). This class will focus more on abstract properties as well as their geometric interpretations, and less on computational aspects. In addition, we will emphasize learning to write clear and compelling proofs.

Math 113 is appropriate for students who have already seen some linear algebra (e.g., Math 51), but linear algebra is not a strict prerequisite.

Depending on your interests, you may prefer Math 104, which emphasizes applications of linear algebra over the abstract theory.

### Textbook and supplementary notes

Our course will primarily use Sheldon Axler's Linear Algebra Done Right.

For further reference, you may also check Professor Katznelson's lecture notes, available here.

Here are some important supplementary notes:

• The tensor product of two vector spaces (after HW 7)
• Wedge products and determinants (after tensor products), including basis-independent definitions of the determinant and characteristic polynomial
• Understanding the cross product in R^3 in terms of the language of wedge products (Note: this will not be on the final at all, and is strictly for those who are interested).

The course grade will be based on the following:

• 30% Homework assignments,
• 30% Midterm exam,
• 40% Final exam

### Homework Assignments

Homeworks will be posted here on an ongoing basis (roughly a week before they are due) and will be due at 4pm on the date listed, at our Course Assistant, Graham White's office (380-R). Please submit your homework either directly to Graham if he is in his office, or slide the homework under his door if he is away. Late homeworks will not be accepted. In order to accomodate exceptional situations such as serious illness, your lowest homework score will be dropped at the end of the quarter. You are encouraged to discuss problems with each other, but you must work on your own when you write down solutions. The Honor Code applies to this and all other written aspects of the course.

Homework problems will often consist of proofs. Even if a problem doesn't start with "Prove that..." you are expected to prove your argument is correct. Please make sure to write complete sentences, not just a string of formulas, and emphasize which axioms, propositions, lemmas, and theorems you use in your argument.

Due date Assignment
Apr 12 Homework 1. Solutions.
Apr 19 Homework 2. Solutions.
Apr 26 Homework 3. Solutions.
May 3 Homework 4. Solutions.
May 10 Homework 5. Solutions.
May 17 Homework 6. Solutions.
May 24 Homework 7. Solutions.
May 30 Homework 8. Solutions.
N/A Homework 9 (suggested problems, not for credit).

### Lecture Plan

Lecture topics by day will be posted on an ongoing basis below. Future topics are tentative and will be adjusted as necessary.

Week Lecture topics Book chapters Remarks
Week 1: Apr 2 - 4 Overview. Fields (real and complex numbers are special cases), and the notion of a vector space over a field. Subspaces. Sums and direct sums of subspaces. Criteria for a sum to be a direct sum. Examples. Chapter 1
Week 2: Apr 9-11 Linear combinations and spans. Finite dimensional vector spaces. Linear independence and the notion of basis. More on basis and dimension. Chapter 2
Week 3: Apr 16-18 Linear maps and isomorphisms. Kernel (null space) and range (image). The matrix of a linear transformation. The vector space of linear maps. Quotient vector spaces, and the rank-nullity theorem. Chapter 3
Week 4: Apr 23-25 Invariant subspaces. Eigenvectors and eigenvalues. Proof that a linear operator on a complex vector space always has an eigenvalue. Diagonal and upper triangular matrices, and invariant subspaces of real vector spaces. Chapter 5
Week 5: Apr 30-May 2 The structure of operators on complex linear vector spaces. Generalized eigenspaces, and nilpotent operators. Nilpotent operators, characteristic polynomial, and the decomposition of any complex finite-dimensional vector space into generalized eigenspaces. An analysis of nilpotent operators. Chapter 8
Week 6: May 7-9 More about the structure of linear maps on complex vector spaces: Jordan normal form. The minimal polynomial, and relation to characteristic polynomial. Analogous (but weaker) results for linear maps on real vector spaces: existence of 1 or 2 dimensional invariant subspaces, block upper triangular matrices, the characteristic polynomial of 2x2 matrices. Eigenpairs, and generalized eigenpair-spaces. Decomposition into generalized eigenspaces and eigen-pair spaces. Chapters 8, 9
Week 7: May 14-16 An introduction to inner product spaces (a generalization of the dot product on Euclidean space). Orthogonal vectors, and orthonormal lists/bases. The Gram-Schmidt procedure for producing orthonormal lists. Orthogonal complements and projections, and the relation to minimization problems. Representing linear functionals by vectors, and the adjoint of a linear map between inner product spaces. Chapter 6
Week 8: May 21-23 More properties of the adjoint of a linear map between inner product spaces (behavior with respect to composition, sums and scalar products, kernels and images). Self-adjoint operators: first properties. Normal operators (a larger class than self-adjoint), and their characterization. The complex and real spectral theorems. Chapters 6, 7
Week 9: May 28-30 Beyond the spectral theorem: normal operators on real vector spaces. The geometry of linear transformations: isometries (length preserving maps). Positive linear operators and the polar decomposition (an analogue of "polar coordinates" for complex numbers). Singular value decomposition (SVD). A brief introduction to signed areas and volumes, via alternating multilinear maps. Chapter 7
Week 10: June 4 Wedge (a.k.a. exterior) products, and the relationship to signed volumes and alternating multilinear maps. Determinants of linear operators constructed as the induced map on the top exterior product of a finite-dimensional vector space. Determinants as scaling signed volume. Sample computations and applications. The characteristic polynomial revisited. Notes above

### Midterm and Final Exam

The midterm was held Tuesday, May 7th, from 7pm - 9pm in room 370-370. Here is a copy of the midterm. Here are solutions.

The topics for the midterm were the first four and a half weeks of class: topics will range from first definitions of vector spaces to linear maps, eigenvalues, eigenvectors. (Chapters 1-5 of the book, not including Chapter 4, aside from the statement of the fundamental theorem of algebra). It was a closed book, closed note exam.

Here is an exam from 2010 that you may have used for practice. Here is a slightly newer one, along with a set of solutions. Bear in mind that the topics cover differ somewhat year to year, so the questions will not be exactly representative of the material you have or have not learned to date.

The final exam was be held on June 11th (Tuesday) from 3.30-6.30pm, in room 380-380Y. Here is a copy of the final. Here are solutions.

The topics for the final were the content of the entire class, with an emphasis on material after the midterm (Chapters 8-9, and 6-7 from Axler, along with class notes on tensor products and wedge products). Here is one practice final from 2010, and here is another one from 2011.