• First week's homework: The first homework assignment is section 1.1 problems 3bd, 6ag, and 1.2 problems 2e, 3d, 5aj, 7,9, 15, 22a. (Here, bd means b and d but not c, and so forth.)
• Notes on the first lecture PDF file..
• Notes on the second lecture PDF Friday Sept 1 lecture overview .
• Notes on Sept 4 lecture. Also the next homework is included. Sept 4 PDF file.
• Notes on Sept 6 class. Problems are stated and then solutions. Sept 6 PDF problems and solutions file.
• Notes on Sept 8 class. Sept 8 PDF file.
• I left out mention of upper and lower triangular. You do have a textbook and I can't improve on it with respect to upper and lower triangular, except to note that there are matrices that cannot be put into LU form. The matrix with 1s in position 1,2 and 2,1 and 0s in positions 1,1 and 2,2 is an example. But these are exceptional cases.
• Homework for Sept 18 is p 94 numbers 1, 3g [do it by cofactors], 3h [do it by row operations], 6, and 9. But with 9, you may use either the cofactors definition from the text of the permutations-based definition from lecture. On page 101, numbers 2, 4, 10, 11, 12. And on page 109, numbers 2d and 3.
• Notes for Sept 11. Determinants and some properties.
• Homework is now listed separately; that should make it easier to find.
• Notes for Sept 13. Permutations, cofactors, and all that jazz.
• Notes for Sept 15. [Updated to correct amputation of solution pages] Assorted problems and solutions.
• Homework for Sept 25 (not 18th!) is section 3.1 numbers 1, 8, 9, 12, section 3.2 [not 2.2] numbers 2, 3, 8, 11, 19, 20 (modified). Pages are 122 and 131-132 respectively.
• Notes for Sept 18. New topic, some key definitions. vector spaces defined.
• Notes for Sept 20. More definitions. null space, row space, span, etc. defined.
• Notes for Sept 22. Exam 1 is on Sept 29. While we will be looking at new material next week, at least in part, the topics for the exam end with chapter 3 section 3. We've covered all of that apart from the \emph{Wronskian}, which also will not be on (this) test. Some of the exam problems will be mainly a matter of terminology. Some will be mainly a matter of computing. You can expect to be given a matrix, for example, and find its inverse or row-reduce it. You can expect to be given a row-reduced augmented matrix and find the corresponding solution set. You can be expected to be asked to compute a determinant. And, some of the questions will be in the spirit of the ones we've been working through on Fridays. Here's Friday Sept 22 problems and solutions.
• Notes for Sept 25. Basis and dimension. read all about it. Also some discussion of how this fits in with the very beginnings of the course and solution sets of systems of linear equations.
• Notes for Sept 27. Coefficient vector; sum and intersection of vector spaces. link here.
• Reminder: Exam is Friday. Bring your own paper and writing instruments. No calculators.
• Homework for [not Monday but Wednesday October 4] Section 3.3 p 143 1ce, 5,6,7,17,20. Section 3.4 p 149 5 and 11.
• Homework for [not Monday but Wednesday Oct 11] Section 3.4 # 17, 18, Section 3.5 # 1a, 2a,5a(iii), 10. Section 3.6 # 1b, 3, 9, 10.
• Synopsis of Friday Oct 6 problems session with solutions. .
• Solutions to Exam 1. As noted, the actual exams had fewer questions each; material has been merged. PDF file.Long-winded, very thorough solutions are good for distribution but not expected as exam responses.
• Synopsis of Oct 9 class definitions reviewed, with applications.
• Homework for Monday Oct 16: Chapter 4, section 1 page 182 etc.: numbers 2, 4, 8, 10, 15, 17, 20. Chapter 4 section 2 p 195 numbers 3b, 5, 9. Chapter 4 section 3 p 202 etc., numbers 1ae, 3, 7, 8, 13, 14, 15. Due following Monday Oct 23. (edit)
• Synopsis of Oct 16 class linear transformations and matrices. .
• Synopsis of Oct 18 class, first draft: similarity. First peek at orthogonality. .
• Homework for this week, due Monday Oct 30: p 224 section 5.1, #3d, 5, 8b, 10, 18. Page 233 section 5.2, #1c, 4, 9, 11, 14. Page 243 # 1c, 2c, 9, 13.
• Synopsis and backfilled synopses for Oct 25 and previous: least squares and underlying theory. .
• Exam 3 will be on Nov 3 as previously provisionally scheduled. Material will be everything we've seen so far plus one topic not yet seen, the Gram-Schmidt algorithm. This algorithm takes as input an ordered basis a_1,a_2,etc. of some subspace (can be the whole thing) of real n-space, and returns a basis of mutually perpendicular vectors b_1,b_2, etc. so that the span of the first j b's is equal to the span of the first j a's. If you like, you can divide each of your b's by its own length to arrive at a basis of mutually perpendicular UNIT vectors g_1,g_2, etc. so that the span of the first j g's is equal to the span of the first j original a's. It's all about projections. Hear all about it Friday.
• (EDITED, NEW FILES) The linked Mathematica and PDF files are executable, and just readable, files that illustrate the Gram-Schmidt algorithm in two contexts; one where an inner product is used and the other where good old dot products are used. (with permissions!) Mathematica Gram-Schmidt code and PDF version of the code, not executable . Also, synopsis of Friday Oct 27.
• Exam is this coming Friday Nov 3, just a reminder. Now, synopsis and elaboration of Oct 30 class.
• Nov 6 (musical) notes here New: notes on the web page example and the Perron-Frobenius theorem: PDF version and Mathematica notebook.
• .
• Solutions to Exam 2, in comprehensive and possibly overly thorough form. All questions from both versions included. PDF file.
• Homework for Wed Nov 15: p 475 numbers 2, 5, 8, 13, 15, 17, 20, 23, 24, 39. P 483 number 44, p 502 number 1, 2, 6, 15, 26.
• More synopses. polar, cylindrical, spherical coordinates. Local unit vector frameworks. Parameterizations, arc length, path integrals. Vector fields PDF version and Mathematica executable notebook.
• Homework due this coming MONDAY Nov 20: p 521 numbers 8, 39. page 528 numbers 1, 2, 23, 24. page 537 numbers 3, 7, 8, 12.
• Notes for Wednesday and Friday: Lots of stuff. Gradient, divergence, curl, and some multiple integral stuff. Polar coordinate conversions. Eigenvectors revisited. Polar coordinates ideas taken into 4 dimensions. Volume of a four-D unit ball.
• Notes for Nov 20. Multiple integrals and path integrals. Path independence and when and where you have it.
• Link to an exam given in a related class by my office roommate. There's a lot of overlap of material between the courses, and we've not seen exam questions over eigenvalues and eigenvectors. So here you are, a look at what is considered to be of central importance. Of course, next semester the questions would be different. With the kind permission of Prof. Borosh.
• Synopsis for Nov 27 class. Divergence theorem and Stokes' theorem.. Green's theorem is just the special case of Stokes' theorem in which the curve and the surface lie flat in the $x,y$ plane.
• Solutions to one version of the third exam linked here. and the other linked HERE.
• Some handy tricks for quicker evaluation of characteristic polynomials of 2 by 2 and 3 by 3 matrices. Plus remarks on how it's done for big matrices. please read this.
• Exhortation and invitation: This is going to be hard. It can't help but be hard. But there is a huge gap between hard and not doable. You-all can do this. Stay with it, accept that at times you'll be stuck and frustrated. Persist. Get help. Ask me questions, during or after class or in office hours. Understanding will come.