### Matrix Theory, Fall 2013

Eigenmural

© 2013 Laure Bukh

Used with permission

#### When:

Mondays, Wednesdays, Fridays 11:30 (lectures), Tuesdays 1:30pm (recitation)#### Where:

Wean Hall 5421 (lectures), Wean Hall 8220 (recitation)#### What:

Linear functions are the simplest functions, and linear equations are the simplest equations. This is a course about them.

We will master simple, yet powerful ways to manipulate the linear functions, and will have fun iterating them to infinity. We learn how to solve linear equations, and discover how to think of dimensions higher than 3.

#### Resources:

The book for the course is *Linear Algebra Done Wrong* by Sergei Treil. It is freely available on author's website. There is a copy hosted by our library.

Not all topics that we cover are in the book, and some topics we will cover differently.

Links to additional resources will be posted as the course progresses.

#### More fun:

More fun can be had at my *office hours* on Mondays 3:30–4:30pm and Thursdays 9:30–10:30am in Wean 6202 or 6th floor lounge. I am also available by appointment.

Jacob Davis (who is our TA) also has office hours. They are Tuesdays 16:30–18:30 and Thursdays 19:00–20:00 in Wean 7112.

#### Course activities:

Mastery of any subject requires practice. Hence, there will be regular homeworks. You are strongly encouraged to do homework individually. Collaboration and use of external sources are permitted, but discouraged, and must be fully acknowledged and cited. Collaboration may involve only discussion; all the writing must be done individually. The homeworks will be returned one week after they are due.

Students are expected to fully participate in the class. Discussions during the lectures are encouraged.

The homework will count for 10% of the grade. During the semester there will be three tests (on **September 18**, **October 16**, **November 25**).
Each test will count for 20% of the grade. The final exam will count for 30%. The date of the final exam has been set by the Registrar to **5:30pm on December 9** in **Doherty Hall 1112**.

Make-up tests will be given only in the case of a documented medical excuse, a university-sanctioned absence, or a family emergency. However, this must be requested **before** the official start time of each test.

#### Lectures:

- August 26: Introduction. Geometric motivation for vectors. Definition of a vector space. Uniqueness of additive inverse.
- August 28: Examples of vectors spaces. Fields. Word “tableau”. Homework #1
- August 30: Bases. Spanning sets. Linearly independent sets. Bases are spanning linearly independent sets. Spanning set contains a basis (first half).
- September 2:
*Labor day* - September 4: Spanning set contains a basis (second half). Infinite-dimensional vector spaces. Linear transformations and matrices. Matrix-vector multiplication. Homework #2
- September 6: Matrix-matrix multiplication. Properties of the multiplication. Matrix transpose.
- September 9: Trace. Trace of a product. Left- and right-invertibility. Isomorphisms.
- September 11: Subspaces. Homogeneous coordinates. Faster matrix multiplication. Faster integer multiplication. Notes on faster multiplication. Homework #3
- September 13: Linear equations. Echelon form. Elementary matrices. Reduced echelon form.
- September 16: Uniqueness and existence of solutions to linear equations. Linearly independent sets and spanning sets in
**R**^{n}. Homework #4 - September 18:
**Test** - September 20: Dimension. Square matrices and invertibility. Range and kernel.
- September 23: Span. Column space. Computing the kernel. Computing range (part I).
- September 25: Computing range (part II). Row space. Rank. Rank-nullity theorem. Number of lines determined by
*n*noncollinear points in the plane. Homework #5 - September 27: Linear transformations in different bases. Change of basis. Similar matrices. Motivation for the determinant.
- September 30: Determinant. Properties of the determinant. Determinant of an upper triangular matrix.
- October 2: Determinant of any matrix. Determinant of the product. Multilinear expansion of the determinant. Homework #6
- October 4: Existence of the determinant. Permutations. Sign of a permutation.
- October 7: Cofactor expansion.
- October 9: Eigenvalues, eigenvectors, eigenspaces. Characteristic polynomial. Fundamental theorem of algebra (statement only). Homework #7
- October 11: Leading and trailing coefficients of a characteristic polynomial. Product of eigenvalues. Algebraic and geometric multiplicity.
- October 14: Eigenvalues and trace. Diagonalizability. Homework #8
- October 16:
**Test** - October 18:
*Mid-semester break* - October 21: Linear independence of eigenspaces. Direct sums.
- October 23: Inner product spaces. Basic properties of the inner product. Homework #9
- October 25: Cauchy–Schwarz inequality. Orthogonality.
- October 28: Orthogonal projections. Gram–Schmidt orthogonalization.
- October 30: Adjoint. Isometry.
- November 1: Unitary transformations. Rigid motions (part 1).
- November 4: Rigid motions (part 2). Upper triangulization. Homework #10
- November 6: Cayley–Hamilton theorem. Addition of algebraic integers. Normal operators. Notes on addition of algebraic integers.
- November 8: Normal operators (cont'd). Hermitian square. Self-adjoint operators. Positive definite operators.
- November 11: Square roots. Singular value decomposition. Matrix presentation of singular value decomposition.
- November 13: Singular value decomposition as a diagonalization. Geometric meaning of the singular value decomposition. Homework #11
- November 15: Distances in Euclidean spaces.
*Guest lecture by Jie Ma* - November 18: Structure of orthogonal operators (part I).
- November 20: Structure of orthogonal operators (part II).
- November 22: Invariant subspaces. Generalized eigenspaces. Homework #12
- November 25:
**Test** - November 27:
*Thanksgiving (part I)* - November 29:
*Thanksgiving (part II)* - December 2: Generalized eigenspaces are linearly independent.
- December 4: Generalized eigenspaces are spanning.
- December 6: Jordan normal form.
- December 9:
**Final exam**is at 5:30pm in Doherty Hall 1112.