Math 3051-001 Theoretical Linear Algebra. Spring 2010

TR 11:00-12:20. Barton Hall Classroom 405.
F 12:00 PM - 12:50 PM (recitation). Barton Hall Classroom 401.

Syllabus

This is a one semester course in linear algebra with a higher degree of abstraction than a traditional undergraduate linear algebra course. Topics include vector spaces, linear transformations, determinants, eigenvalues and eigenvectors, canonical forms, and inner product spaces.

Textbook:
Sheldon Axler, Linear Algebra Done Right, second edition, Springer, 1997. See the book website for more information including errata.

For any questions, contact the instructor, Prof. Daniel B. Szyld, at 215.204.7288 (szyld AT temple DOT edu); or the Teaching Assistant Zhiyong Feng at (zhfeng AT temple DOT edu).

What we covered in class:
Tue. 19 Jan. Chapter 1: Definition of vector space, subspaces, examples, intersection and sum of subspaces.
Thu. 21 Jan. Direct sums of subspaces (finished Chapter 1). Span and linear independence (starting Chapter 2).
First Homework: Exercises 6, 7, 9, 10, 12, 13, 15 (pages 19-20), due Tuesday 26 January.
Tue. Jan 26. Linear independence lemma. Theorem on number of linear independent vectors.
Thu. Jan 28. Basis. Dimension.
Second Homework: Chapter 2 exercises 3, 9, 11, 12, 13, 15 (pages 35-36). Due Tuesday 2 February.
Tue. Feb 2. Dimnension of sum and direct sum of subspaces. Linear transformations. Space of linear transformations.
Thu. Feb. 4. Null space and range. Injective and Surjective linear transformations.
Third Homework: Chapter 3, exercises 4, 5, 10, 22, 23 (pages 59-61). Due Tuesday 9 February.
Tue. Feb. 9. Fundamental theorem on the dimentions of the null space and range. Matrix of a linear transormation.
Thu. Feb. 11. (Snow day - no class).
Tue. Feb. 16. Invertibility of a linear transformation. Quiz.
Thu. Feb. 18. Invariant subspaces. Eignevalues and eigenvectors.
Fourth Homework: Chapter 5 (part I), exercises 1, 3, 9, 10, 11, 12, 14 (pages 94-95). Due Tuesday 23 February.
Tue. Feb. 23. Lineraly independent eigenvecotrs and diagonal representation. Triangular representation and inveriant subspaces.
Fifth Homework: Chapter 5 (part II), exercises 18, 19, 20, 21 (page 95). Due Thursday February 25.
Thu. Feb. 25. Triangular representation and invertibility.
Tue. March 2. Inner product. Norms,
Thu. March 4. Midterm.
Tue. March 16. More on inner products, orthogonal projections. Orthonormal bases. Gram-Shmidt orthonormalization process.
Homework 6, due Tuesday March 23.
Thu. March 18. Examples of Gram-Schmidt. Orthogonal polynomials. QR decomposition of a matrix.
Tue. March 23, Schur decomposition anf Schur form. Minimium distance.
Thu. March 25, Linear Functional. Representation theorem.
Homework 7: Chapter 6 (part II) exercises 26-32 (page 125). Due Thursday 1 April.
Tue. March 30, More on norms of operators. Self-adjoint and normal operators. Spectral theorem.
Homework 8, due Tuesday April 6.
Thu. April 1, Isometries.
Tue. April 6. More on normal operators.
Ninth Homework: Chapter 7, exercises 1-8 (pages 158-159). Due Thursday April 8.
Thu. April 8. Positive Operators. Square Root. Polar Decomposition.
Tenth Homework: Chapter 7, exercises 16-18, 20-22, (pages 159-160). Due Tuesday April 13.
Tue. April 13. More Polar Decomposition. Singular Values.
Eleventh Homework: Chapter 7, exercises 26, 28-32 (pages 160-161). Due Thursday April 15.
Thu. April 15. Exam
Homework 12, due Tuesday April 20.

Coming up: Chapter 8.