Sat Aug 19 11:12pm: This class was a lot of fun. Unfortunately, I don't think I did very well on the tests, but the material was very interesting and you teach it very well. Thanks for all your help throughout the summer!
Thu Aug 17 4:18pm: Hi Santi, I was wondering if the HW solutions for 10.4 and 10.5 will be posted before the final? Thanks
Reply: Unfortunately no. However, the practice final from last summer and final have questions that come from those sections, so you can try to use those as a guide. Please email me with any specific questions you may have.
Wed Aug 16 6:50pm: Hi Santi, do we need to know all 3 ways of finding the special fundamental matrix, or is one enough?
Reply: You can skip the method where you solve for each column (by solving for initial conditions) separately. But, you should be able to compute Phi(t) using the two other ways -- either by using any fundamental matrix, or by computing e^(At).
Wed Aug 9 2:07pm: Hi Santi, Would it be possible for you to post solutions to the final you gave last summer?
Reply: Ah yes, that would be good :) Look for them this weekend or possibly sooner.
Reply #2: They are now up on my site from last summer. Enjoy :)
Wed Aug 9 2:03pm: Hey Santi, the final covers lecture up til Wednesday, right?
Reply: Yep, up to and including 10.5.
Sun Jul 30 12:52pm: Hi Santi, Does the midterm cover material up to 5.3 or 5.4?
Reply: The midterm covers everything from 3.7 through 5.4 of the linear algebra we covered, AND 3.2 and 3.3 in the differential equations book ;)
Fri Jul 28 7:11pm: Hi Santi, I've been consistently checking my answers with answers in the back of the book, but somehow some of the eigenvectors associated with the same eigenvalue of symmetric matrices are different in the answers than what my work yields. However, the eigenvectors are still orthogonal to the eigenvectors of different e-values, and I can still apply the g-s process to obtain an orthogonal matrix Q. What I am asking is, how is it possible that the book obtains some different Q with the same eigenvalues yet the diagonalization yields the same matrix A?
Reply: These diagonalization problems, including for symmetric matrices, can have many correct answers. Since there are many possible eigenvectors associated with an eigenvalue, and many possible orthonormal eigenvectors associated with an eigenvalue of a symmetric matrix, there are many possible matrices S or Q in the diagonalization process. The wonder of it all is that they all give the same matrix A when forming S Lambda S^-1 (or Q Lambda Q^T)! The book may have done Gram-Schmidt on eigenvectors different than the ones you use, thus getting a different matrix Q. As you said, as long as the vectors you use are in fact eigenvectors and are orthogonal, you can still apply G-S and get an orthogonal matrix Q that works. When in doubt, ask me about a specific answer you come up with.
Wed Jul 26 8:31am: Hi, I found the material on quiz six pretty difficult. Could you put the solutions up? Thanks.
Reply: That quiz was indeed harder than usual. The solutions will be up later today.
Tue Jul 25 9:54am: Hi Santi, Is there a quick way to check if our answer to a 5x5 determinant is correct?
Reply: Good question. Probably the best way is to compute the same determinant using a cofact expansion along a different row or column than the one you used, and see if you get the same result (you should). There's really no other quick way of checking, other than asking me if your answer is correct ;)
Mon Jul 17 1:56am: Hi, Santi, i found out midterm is so different from our previous homework and quizzes. Actually, it is much harder. i have fully reviewed all the materials before the midterm. but it turn out that i can not handle some derivative changes of those materials as they appear in the midterm. Is our second midterm gonna be as hard as this one? what can i do to improve my grades.
Reply: I do agree that this midterm was somewhat more difficult than the homeworks and quizzes (as midterms usually are), but not too much. The second midterm may be a bit easier, but then again the material is a bit harder so it kind of evens out :) As I said before, if you didn't do too well, don't worry too much about it. If you show improvement in the rest of the course, I definitely take that into consideration when computing grades. So, maybe the first midterm you should just think of as a wake up call. If you are having trouble, come to office hours and ask questions. One not-so-good score won't ruin your grade, but you should definitely make sure you do well on the quizzes and exams that are left. Please let me know if you have any other concerns.
Thu Jul 13 8:37pm: if we are trying to answer the question of whether certain vectors form a basis for a region, is it ok to prove that the vectors are linearly independant, and then simply state that "since the vectors are lin. indep., then they automatically span the region" so the vectors are a basis?
Reply: This only works when you already know the dimension of the space and there are as many vectors as that dimension. For example, if you have three linearly independent vectors in R^3, then you can say that they automatically span and so form a basis since R^3 has dimension 3. However, if you don't know the dimension of the space beforehand, you would indeed have to show that the vectors span the space, in addition to showing they are linearly independent.
Wed Jul 12 11:02pm: if the last entry of a matrix is the first non-zero number in that row, is that a pivot also?
Reply: Yep, a pivot is just the first nonzero entry in a row; whether that entry occurs as the last entry of the matrix or not doesn't matter.
Tue Jul 4 9:55am: Hi. I was wondering if you were planning on GSI-ing for this class in the fall?
Reply: Actually I won't be teaching at all next year. But, this website and my previous websites will stay up, so you can use any information on here that you find useful.
Tue Jun 27 6:14pm: will the reduced echelon form be on the quiz?
Reply: Nope, just good ole regular echelon form. We won't talk about reduced echelon form again, but it's a good thing to know about.
Fri Jun 23 3:32pm: This is a test comment.