# Math 110

## Linear Algebra

Lectures TuTh 14:00-15:30, 100 Lewis

Office hours TuTh 11:00-12:00, 821 Evans Hall

### Table of Contents

#### Topics covered

• 08/29: Vector spaces. Subspaces.
• 08/31: Linear combinations of vectors. Linear dependence and independence.
• 09/05: More on linear (in)dependence. Bases and dimension.
• 09/07: Examples of bases. Linear transformations.
• 09/12: Ranges and kernels. The rank and nullity formula.
• 09/14: The matrix representation of a linear map. Composition of maps and matrix multiplication.
• 09/19: Applications of matrix theory to graph theory. Invertibility of linear maps and matrices.
• 09/21: Isomorphisms.
• 09/26: Changes of coordinates.
• 09/28: Dual spaces and dual bases.
• 10/03: Duality and linear maps. Differential operators.
• 10/05: Solving constant-coefficient ODEs.
• 10/10: Elementary row/column operations and matrices.
• 10/12: Rank-revealing forms; reduced row echelon form.
• 10/17: Midterm.
• 10/19: System of linear equations: theoretical and computational aspects.
• 10/24: Determinants of order 2 and 3; recursive definition for all n.
• 10/26: Determinants of order n. Properties of determinants.
• 10/31: Eigenvalues and eigenvectors.
• 11/02: Diagonalizability.
• 11/07: Markov chains.
• 11/09: Invariant subspaces and the Cayley-Hamilton theorem.
• 11/14: The Jordan normal form (over C).
• 11/16: Applications of the Jordan normal form. The minimal polynomial.
• 11/21: Introduction to MATLAB.
• 11/28: Inner products and norms.
• 11/30: Gram-Schmidt orthogonalization.
• 12/05: The adjoint of a linear map. Normal and self-adjoint maps.
• 12/07: Unitary and orthogonal maps. The spectral theorem.

### Resources

• Textbook: S. H. Friedberg, A. J. Insel, L. E. Spence Linear Algebra, 4th ed.
• Carl de Boor's online notes on applied linear algebra.
• Some papers related to the Jordan normal form: Vlastimil Ptak's, Carl de Boor's, O.H.'s.
• Other books:
• R. Horn and C. Johnson Matrix Analysis, Topics in Matrix Analysis,
• P. Halmos Finite-dimensional vector spaces,
• S. Axler Linear Algebra Done Right.

### Academic honesty

The instructor welcomes cooperation among students and the use of books. However, handing in homework that makes use of other people's work (be it from a fellow student, a book or paper, or whatever) without explicit acknowledgement is considered academic misconduct.

### Assignments

All homework problems are from the 4th edition of Friedberg et al.

• Homework assignment #1, due Wed Sep 6th:
Sec 1.2 - 1, 8, 10, 22 (read appendix C); Sec 1.3 -- 1, 3, 9, 13, 30.
• Homework assignment #2, due Wed Sep 13th:
Sec 1.4 - 4, 15; Sec 1.5 - 1, 3, 17; Sec 1.6 - 1, 8, 13, 21, 26.
• Homework assignment #3, due Wed Sep 20th:
Sec 2.1 - 10, 15, 28, 35; Sec 2.2 - 3, 10, 13; Sec 2.3 - 2, 3.
• Homework assignment #4, due Wed Sep 27th:
Sec 2.3 - 13, 15, 17, 20; Sec 2.4 - 1, 2, 6, 17, 22.
• Homework assignment #5, due Wed Oct 4th:
Sec 2.5 - 3, 6, 7, 10; Sec 2.6 - 1, 2, 4, 6, 14, 15.
• Homework assignment #6, due Wed Oct 11th:
Sec 2.7 - 1, 2, 4, 9, 12, 14, 16, 17, 18.
• Homework assignment #7, due Wed Oct 25th:
Sec 3.1 - 3, 5; Sec 3.2 - 2, 6, 8, 18; Sec 3.3 - 1, 11; Sec 3.4 - 1, 2.
• Homework assignment #8, due Wed Nov 1st:
Sec 4.1 - 1, 2, 4, 10; Sec 4.2 - 1, 2, 6, 8, 25, 26.
• Homework assignment #9, due Wed Nov 8th:
Sec 4.3 - 10, 11, 14, 15, 23; Sec 4.4 - 5; Sec 4.5 - 1, 4, 6, 10.
• Homework assignment #10, due Wed Nov 15th:
Sec 5.1 - 3, 22; Sec 5.2 - 1, 8, 11; Sec 5.3 - 1, 2, 3, 4, 13.
• Homework assignment #11, due Wed Nov 29th:
Sec 5.4 - 1, 4, 5, 17, 18, 42; Sec 7.2 - 1, 4, 6, 14, 19; Sec 7.3 - 1, 2, 8.
• Homework assignment #12, due Wed Dec 6th:
Sec 6.1 - 1, 3, 9, 17; Sec 6.2 - 2ab, 9, 15, 16; Sec 6.3 - 3, 12.
• Suggested practice problems for Sections 6.4 - 6.6:
Sec 6.4 - 9, 11, 13, 19; Sec 6.5 - 2, 8, 15; Sec 6.6 - 1, 4, 7.

### Exams

Midterm: Oct 17, in class. Here is a mock midterm, in PS and in PDF and its solutions in PS and in PDF.

Final: Dec 13, 12:30-3:30, in 237 Hearst Gym. Here is a mock final test, in PS and in PDF.

Review sessions will be held

• by Darren Rhea on Monday from 10am to 2pm with lunch, at 2 LeConte Hall;
• by Emily Peters on Monday from 7pm to 9pm at 85 Evans Hall;
• by Erik Closson on Monday from 3pm to 6pm in 7 Evans Hall and on Tuesday from 1pm to 3pm at 855 Evans Hall;
• by Jeff Brown on Tuesday from 9:30am to 11:30am at 1015 Evans Hall; he also wrote a review sheet, available outside of 1066 Evans Hall.

### Demos, m-files, etc.

Last modified: Dec 11, 2006