Math 110

Linear Algebra

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

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

GSIs: Jeffrey Brown, Eric Closson, Emily Peters, Darren Rhea.

Table of Contents

• Course Info
• Resources (books, notes, more)
• Academic honesty
• Assignments
• Solutions
• Exams
• Demos, mfiles, etc

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.

Solutions

• Solutions to homework assignment #1: in PS and in PDF.
• Solutions to homework assignment #2: in PS and in PDF.
• Solutions to homework assignment #3: in PS and in PDF.
• Solutions to homework assignment #4: in PS and in PDF.
• Solutions to homework assignment #5: in PS and in PDF.
• Solutions to homework assignment #6: in PS and in PDF.
• Solutions to homework assignment #7: in PS and in PDF.
• Solutions to homework assignment #8: in PS and in PDF.
• Solutions to homework assignment #9: in PS and in PDF.
• Solutions to homework assignment #10: in PS and in PDF.
• Solutions to homework assignment #11: in PS and in PDF.
• Solutions to homework assignment #12 in PS and in PDF.

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.