# Math H110

## Honors Linear Algebra

TuTh 9:30-11:00am, Room 2 Evans Hall

Office hours: 11:00am-12:00noon, Room 821 Evans Hall

Course control number: 54217

### Syllabus

This syllabus is to be taken with a grain of salt: it is to be fine-tuned as we move along.
• 08/23: Fields. Vector spaces. Examples. Comments.
• 08/28: Linear dependence. Linear combinations. Bases. Dimension.
• 08/30: Isomorphism. Subspaces. Calculus of subspaces. Dimension of a subspaces.
• 09/04: Dual spaces. Brackets. Dual bases. Reflexivity.
• 09/06: Annihilators. Direct sums. Dimension of a direct sum. Dual of a direct sum.
• 09/11: Quotient spaces. Dimension of a quotient space. Bilinear forms.
• 09/13: Tensor products. Product bases.
• 09/18: Permutations. Cycles. Parity.
• 09/20: Multilinear forms. Alternating forms. Alternating forms of maximal degree.
• 09/25: Linear transformations. Transformations as vectors. Products. Polynomials.
• 09/27: Inverses. Matrices. Matrices of transformations.
• 10/02: Invariance. Reducibility. Projections. Combinations of projections.
• 10/09: Change of basis. Similarity. Quotient transformations.
• 10/11: Range and nullspace. Rank and nullity. Transformations of rank one.
• 10/16: Tensor products of transformations. Determinants.
• 10/18: Proper values (eigenvalues). Multiplicity. Triangular form.
• 10/23: Nilpotence. Jordan form.
• 10/25: Midterm.
• 10/30: Inner products. Complex inner products. Inner product spaces. Orthogonality.
• 11/01: Completeness. Schwarz's inequality. Complete orthonormal sets.
• 11/06: Projection theorem. Linear functionals. Parentheses versus brackets.
• 11/08: Natural isomorphisms. Self-adjoint transformations. Polarization.
• 11/13: Positive transformations. Isometries. Change of orthonormal basis.
• 11/15: Perpendicular projections. Combinations of perpendicular projections. Complexification.
• 11/20: Characterization of spectra. Spectral theorem.
• 11/27: Normal transformations. Orthogonal transformations. Functions of transformations.
• 11/29: Polar decomposition. Commutativity. Self-adjoint transformations of rank one.

### Resources

Required Text:
• Halmos, Paul R. Finite-dimensional vector spaces. Reprinting of the 1958 second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1974. viii+200 pp.
• Halmos, Paul R. Linear algebra problem book. (English summary) The Dolciani Mathematical Expositions, 16. Mathematical Association of America, Washington, DC, 1995. xiv+336 pp. ISBN: 0-88385-322-1
• Gel'fand, I. M. Lectures on linear algebra. With the collaboration of Z. Ya. Shapiro. Translated from the second Russian edition by A. Shenitzer. Reprint of the 1961 translation. Dover Books on Advanced Mathematics. Dover Publications, Inc., New York, 1989. vi+185 pp. ISBN: 0-486-66082-6
• Glazman, I. M.; Ljubic, Ju. I. Finite-dimensional linear analysis. A systematic presentation in problem form. Translated from the Russian and edited by G. P. Barker and G. Kuerti. Reprint of the 1974 edition. Dover Publications, Inc., Mineola, NY, 2006. xx+520 pp. ISBN: 0-486-45332-4
Math H110 on Piazza

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

Homework is assigned every week, due one week later. In solving homework problems, you may use any result in the book that precedes the problem in question, including results of preceding homework problems. It is not OK to use any material following the problem. Problems below are taken from the main textbook for this class and are numbered x.y where x is the section number, y the problem number within the section.
• Homework #1 due August 30th: read Sections 1-4, solve problems 1.1, 1.2, 1.3, 1.4, 4.1, 4.2, 4.4, 4.5.
• Homework #2 due September 6th: read Sections 5-12, solve problems 7.2, 7.3, 7.7, 7.9, 9.1, 9.3, 12.4, 12.7.
• Homework #3 due September 13th: read Sections 13-20, solve problems 14.3, 14.6, 17.1, 17.2, 17.3, 17.5, 20.1, 20.5.
• Homework #4 due September 20th: read Sections 21-25, solve problems 22.1, 22.4, 22.5, 23.2, 23.4, 23.5, 25.4, 25.6.
• Homework #5 due September 27th: read Sections 26-31, solve problems 27.6, 27.8, 27.10, 28.1, 28.2, 31.2, 31.3, 31.4.
• Homework #6 due October 4th: read Sections 32-38, solve problems 33.1, 33.5, 35.2, 36.3, 36.7, 38.11, 38.14, 38.16.
• Homework #7 due October 11th: read Sections 39-45, solve problems 40.1, 40.2, 40.3, 43.2, 43.3, 43.5, 43.6, 43.7.
• Homework #8 due October 18th: read Sections 46-51, solve problems 47.3, 47.5, 49.3, 49.5, 51.3, 51.5, 51.7, 51.9.
• Homework #9 due October 25th: read Sections 52-58, solve problems 52.3, 53.11, 55.1, 58.15.
• Homework *** due November 1st: read supplementary material on the Jordan normal form, write your own proof of the Jordan normal form, ideally with a running example. The end result should be concise but/and self-contained.
• Homework #10 due November 8th: read Sections 59-65, solve problems 62.2, 62.3, 62.4(a)-(c), 65.1, 65.2, 65.3, 65.5, 65.6(a).
• Homework #11 due November 15th: read Sections 66-72, solve problems 69.1, 69.3, 69.7, 70.1, 70.3, 70.9, 72.4, 72.7.
• Homework #12 due November 29th: read Sections 73-79, solve problems 74.1, 74.6, 74.11, 76.2, 76.4, 77.3, 78.3, 79.3.
• To do before the final test: read Sections 80-85, (optional) think about problems 80.1, 80.2, 80.5, 81.1, 82.1, 82.8, 82.9, 84.3, 84.5, 85.2.

### Misc. handouts

• Mock midterm test can be found here, in PS and in PDF.
• Mock final can be found here, in PS and in PDF.
• Vlastimil Ptak's article "A remark on the Jordan normal form of matrices".
• Carl de Boor's article "On Ptak's derivation of the Jordan normal form".