21 B
Mathematics Math21b Spring 2008
Linear Algebra and Differential Equations
Course Head: Oliver Knill
Office: SciCtr 434
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Question:What kind of partial differential quations can we solve using Fourier? Answer: Any linear equation of the form ut = p(D2) or utt = p(D2) where p is a polynomial. In the first case, define ln = p(-n2). The solution is
 u(t,x) = sumn bn exp(lnt) sin(n x) 
where bn are the Fourier coefficients of the initial condition u(0,x). Similar in the wave equation part. If you want to dig further (we do not cover this in this course), you can even solve systems of the form ut = p(D2) + f(t) or utt = p(D2) + f(t). We solve then the homogeneous case as before and use a "cookbook" or operator method find one solution of the inhomogeneous equations for one of the Fourier modes.
Question:When solving for a closed solution to dynamical system questions, what happens if we cannot form an eigenbasis? Answer: Yes, there is a general method which allows to solve such systems but it is beyond this course. The theorem to use is called the Normal form theorem. It allows to find a matrix in which the matrix A restricted to an eigenspace is a "shear" like in
    d/dt x   = x + y
    d/dt y   = y
This is a non diagaonalizable case. To solve it first solve the second equation y(t) = y(0) et. No solve the first x' - x = y(0) et which has (use the cookbook method) the solution x(0) et + t y(0) et. If you understand this example, you see what is going on in general.
Question:In section 2.4 partitioned matrix are used. Do we have to know about that. Answer: The concept of partitioned matrices allows to do many things faster. You look at some matrices as matrices of matrices. For example, for
   |  2 3    0  0  |     | A  0 |
   |  3 5    0  0  |  =  | 0  B |
   |  0 0    1  2  |
   |  0 0    3  7  |
you can immediately see the inverse
    | A-1  0    |    | 5 -3  0  0 |
    | 0     B-1 | =  | -3 2  0  0 |
                     | 0  0  7 -2 |
                     | 0  0 -3  1 |
by inverting the submatrices. The concept is in general useful only if there are submatrices which are zero. It is a time saver because inverting a 4x4 matrix with Gauss Jordan is slower than doing two inversions of 2x2 matrices and for 2x2 matrices,we can write down the inverse directly: swap the diagonal, negate the side diagonals and divide by the determinant.
Question:For 2.4 number 28, is there a systematic way to solve that or is it mostly guess and check? Or did I only arrive at an answer because I was lucky ? Answer: this was a discovery task. There will be a more systematic way later when we know about other topics where it will be possible to derive this systematically. This result is important to keep in mind. Such matrices even have a name, they are called nilpotent.
Question:Will we be allowed to use calculators during the exam? Answer: No. As in Math21a, we do not allow calculators in Math21b. Consequently, it is a good idea to limit the use of calculators and computer algebra systems when doing homework. It can be a good idea to check work using the calculator, but not to do the problems primarily. You should practice as much as possible to do things by hand. In the first week, row reduction is an important algoririthm to master on paper.
Question: Answer:
Question:Which page should I bookmark, the iSites page or the FAS page? Answer: It is safe to bookmark the FAS page.
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Math21b | Oliver Knill | Spring 2008 | Department of Mathematics | Faculty of Art and Sciences | Harvard University