Dror Bar-Natan: Classes: 2000-01:

# Linear Algebra for Engineering (2)

Instructor: Dror Bar-Natan, drorbn@math.huji.ac.il, 02-658-4187.

Plan: I hope to use Sunday classes for applications, as many as we can fit and digest, and Tuesday classes to learn more linear algebra, as below:

• All about determinants: properties, volume, Cramer's rule and matrix inversion.
• All about cannonical forms of matrices: eigenvalues, eigenvectors, diagonalization, etc.
• All about inner products: motivation, definition, basic properties, orthogonality, Gram-Schmidt, orthogonal projections.
If we run out of time, we may have to take some of Sunday classes for theoretical subjects.

Main textbook: There will be none - but I hope to find somebody who would take good notes and I hope to put them on this web site after every class. That's just one reason why you must have internet access if you're taking this course!

Other books: Amitzur's Algebra A, Shores' world of linear algebra, Anton and Rorres' Elementary Linear Algebra with Applications, Hoffman and Kunze's Linear Algebra and many more.

Classes: Sundays 11:00-12:00 at Kaplan B and Tuesdays 10:00-12:00 at Shprintzak 115.

Office hours: Sundays 14:00-15:00 in my office, Mathematics 309.

Problem Sessions: Mondays 10:00-11:00 and 11:00-12:00 at Shprintzak 29, with Liat Kessler, kessler@math.huji.ac.il.

The grade: The final grade will be a weighted average of the final exam grade f and the homework grade h, with weights 0.85f+0.15h if h>f and 0.93f+0.07h if h<f (i.e., the carrot is bigger than the stick). The responsibility for the homework grade is entirely in the hands of Liat Kessler; I am told that she will base the grade on the best 8 assignments that each of you will submit.

### By the week:

 March 4, 6 CT scanner and scan; Class notes for March 4th (introduction to tomography and re-stating it in linear algebra terms); Class notes for March 6th (theorem about the relevance of volume, properties of 2D area of parallelograms, signed area, the definition of volume forms, permutations, signs of permutations). March 13 March 11 was Purim. Class notes for March 13th (computing the standard area form in the plane, all area forms in the plane are proportional, properties of the sign of permutations, the behaviour of general volume forms under general permutations of the arguments, every permutation is a product of transpositions). March 18, 20 What was the picture? A riddle in tomography: Mathematica Notebook (requires Mathematica or MathReader). PDF (requires Acrobat Reader or GhostView. The data file CTScan.m. Class notes for March 18th (reminder on tomography, a 50 HW points prize offering and some explanation of the programs in the handout, estimation of the computational complexity of Gaussain elimination). Class notes for March 20th (the "fundamental formula" for a volume form evaluated on linear combinations, the definition of detemeninats, examples, vanishing on a basis implies vanishing completely, proportionality of volume forms, existance of volume forms - the needed properties of determinants). Homework assignment #1 (determinants). Solutions. March 25, 27 Algebraic reconstruction techniques (PDF) (no program, just output). Class notes for March 25th (first discussion of the iterative way for solving the tomography problem). Tuesday's class was cancelled, makeup sesssion on May 2nd 8:00-10:00 at Levin 8. Homework assignment #2 (determinants). Solutions. April 15, 17 April 1, 3, 8 and 10 are Passover. Class notes for April 15th (review and proof of the the properties of determinants needed for the existance of volume functions). Homework assignment #3 (determinants). Solutions. Class notes for April 17th (lots of extra properties of determinants, rook arrangements and determinants). April 22, 24 Class notes for April 22nd (the determinant of a product and little about row expansions). Homework assignment #4 (determinants). Solutions. Class notes for April 24th (row and column expansions, the adjoint matrix and a formula for the inverse matrix, Cramer's law). April 29, May 1, 2 Class notes for April 29th (reproduction laws for rabbits, powers of matrices, powers of conjugates of diagonal matrices, eigenvalues and eigenvectors). Homework assignment #5 (the adjoint matrix, Cramer's law, population problems). Solutions. Class notes for May 1st (full diagonalization when there is a basis made of eigenvectors). May 2nd is a makeup session for the lost class from March 27th! At Levin 8, 8:00-10:00. Class notes for May 2nd (complex eigenvalues, repeated eigenvalues). May 6, 8 Class notes for May 6th (the Cayley-Hamilton theorem, the case of repeated eigenvalues). Homework assignment #6 (diagonalization and Cayley-Hamilton). Solutions. Class notes for May 8th (More on repeated eigenvalues, systems of linear differential equations). May 13, 15 Class notes for May 13th (more on linear differential equations, exponentiation in the case of repeated eigenvalues). Homework assignment #7 (complex numbers as matrices). Solutions. Class notes for May 15th (More on repeated eigenvalues, more on differential equations, the Jordan canonical form). May 20 Class notes for May 20th (inner products on R2 and in general). Homework assignment #8 (the Jordan form, inner products) Solutions. May 22 is Yom HaStudent. May 29 May 27 is Shavuot. Class notes for May 29th (complex inner products, the cosines law, Pythagoras's theorem). June 3, 5 Class notes for June 3rd (the Cauchy-Scwartz inequality and the triangle inequality). Homework assignment #9 (inner products). Solutions. Class notes for June 5th (orthonormal bases). June 10, 12 Class notes for June 10th (the Gram-Schmidt orthogonalization procedure). Homework assignment #10 (inner products). Solutions. Class notes for June 12th (Orthogonal matrices and orthogonal diagonalization of symmetric matrices - statement and application to min/max problems in multivariable calculus). June 17, 19 Class notes for June 17th (Algebraic Reconstruction Techniques and tomography). Homework assignment #11 (orthogonal diagonalization of symmetric matrices). Class notes for June 19th (proof of the orthogonal diagonalization theorem for symmetric matrices). Thursday, July 12 We will have a review session at 8 Levin at 10:00AM. (Handout). Tuesday, July 17 The final exam (Moed A) at Ulam Canada from 13:30 until 15:30. (Definition, the thing itself). Sunday, September 9 The final exam (Moed B) at Mathematics 110 from 10:00 until 12:00. (Definition, the thing itself). Thursday, January 31, 2002 The final exam (Moed C). (Definition, the thing itself).

Class Photo, April 17, 2001 Click on the image to get the full size version