Math 341- Summaries of lectures and homework assignments.

1/24: We talked about linear transformations (4.1 and 4.2 of Cullen).  To finish next time: show that a linear transformation is one to one if and only if its null space is 0.  Also finish showing that the vector space L(V,W) is isomorphic to Rmxn and hence has the same dimension.

1/26: 4.2  of Cullen representing a linear transformation as a matrix
1/29: 4.3 of Cullen, composition of linear tyransformations corresponds to multiplying the matrices which represent them.

1/31: 4.3, 4.4 of Cullen, Similar matrices, inverses. 4.5,  characteristic values. Quiz 1 on 4.1 and 4.2.

2/2:  More  characteristic values
2/5: We finished 4.5, in particular show that if the characteristic polynomial has distinct roots then the operator is diagonalizable.  We also looked at a strategy for dealing with complex characteristic vectors, use instead their real and imaginary parts, which are real, in a basis.  This leads to 2x2 blocks on the diagonal which are a rotation and a stretch.  After this we started talking about complex dot products.  The ordinary dot product does not work well (for example nonzero vectors would sometimes have length 0).  So Cullen defines x dot y as x*y where x* is the conjugate transpose of x.  Some authors (and matlab) instead use the formula x dot y = y*x.  This works equally well.  The only difference is one formula gives you the complex conjugate of the answer you get using the other formula.

2/7: Quiz 2 on characteristic vectors and characteristic values etc. (4.5).  We then talked about 4.6, orthogonal and unitary matrices.

2/9: 4.7 Gram-Schmidt process.  We also briefly talked about analogues of the dot product on a general real or complex vector space.  The prime example is the inner product on the vector space of real or complex valued continuous functions with domain [a,b].  We define <f,g> = the integral from a to b of f(t)g(t) dt in the real case and in the complex case, just conjugate f or g.  Everything we do using dot products (orthogonality, projections, Gram-Schmidt, etc) can be done in the same way with inner products.  For example if [a,b] = [0,2 pi] a useful orthogonal set is 1, sin t, cos t, sin 2t, cos 2t, sin 3t, ....., or better yet for computations, 1, eit, e-it, e2it, e-2it,...  which give rise to Fourier series.  Note while this material is normally included in Math 240, it is not in Cullen and I will not test you on it.

2/12: 4.8 Assuming Schur's Theorem we showed that if A is a normal matrix (A*A=AA*) then there is a unitary P so that P*AP is diagonal.  If A is real and symmetric, there is an orthogonal matrix P so that PTAP is diagonal.  We then started to talk about quadratic forms and showed that any quadratic form can be written in the form XTAX for a symmetric matrix A.

2/14: No class, University closed

2/16: Cullen 4.8, Colley 4.1,4.2 We show any quadratic form after a change of orthonormal basis has only square terms.  Thus if q(X) = XTHX is a quadratic form (with H symmetric) then
I then stated without justification the Taylor's formula  f(x) is approximately f(p) + grad f . (x-p) + (x-p)T H(x-p)/2 for x near p. Here H is the matrix of second partial derivatives of f. I then showed that if f has a local max or min at p then the gradient of f is 0 at p.  If the gradient of f is 0 at p and all char values of H are positive then f has a local min at p.  If the gradient of f is 0 at p and all char values of H are negative then f has a local max at p.  If the gradient of f is 0 at p and some char values of H are positive and some are negative but none are 0 then f has a saddle at p.  If some char values of H are 0 you need to go to higher order terms of the Taylor series to understand f (although if some char values are negative and some are postive you know p cannot be a local max or min).

In the interest of time pressure, I'll leave you to read the proof of Schur's theorem in the book rather than present it in class.



2/19: Colley 4.1, 4.2 we showed that in two dimensions there is a shortcut test to classify critical points. Colley has a corresponding test for any dimension but beyond 2, you may as well use matlab for your calculations so why not just find the characteritic values.  Quiz 3 on Gram-Schmidt

2/21: Quiz 4 on Colley 4.2.  After answering some homework problems we talked about the method of Lagrange multipliers 4.3.

2/23: We did a bunch of problems in 4.3.

2/26: Started diff Equations  Braun 1.1, 1.2, 1.4
2/28:  Quiz 5 on Lagrange multipiers.  We looked at an outline of the proof of existence of uniquenes for solutions to the IVP y'=H(y,t), y(t0) = y0 for reasonable H.  We looked at this in a more general situation than that given in the book, where y is vector valued and H only needs to be Lipschitz.   Details are given in a handout.  Braun 1.10 you can ignore the problems about estimating the interval on which an IVP can be solved.

3/2: Exam #1 covering Colley 4.1-4.3 through p. 266, Cullen 4.1-4.8.   You may bring a 3x5 card to the exam.  You may also bring an inexpensive scientific calculator, but not a programmable or graphing calculator.  Here is an old 341 exam 1 with answers.  On testbank there are old 241 and 241H exams with max/min/Lagrange multiplier problems.  Relevant problems for 241 exams  are:
Relevant 241H problems are:
Relevant 461 problems are:
3/5: Braun 1.9   Exact equations.
3/7:  Quiz 6 on first order methods, linear, exact, and separable.  Braun 1.13, 1.15, 1.16   Euler's method, and improved Euler's method, Runge-Kutta methods,

3/9: 2.1, introduction to second order diff equations- existence and uniqueness of IVP, and second order linear homogeneous case, Wronskian.

3/12:  linear second order homogeneous with constant coefficients.  2.2, 2.3 , start of 2.5.

3/14: 2.4, 2.5, 2.6:  Solving nonhomogeneous linear ODEs, Examples of nonhomogeneous ODEs, forced vibrations. Quiz 7 on 2.1-2.3.

3/16: 2.8  A brief overview of solving ODEs by power series.
3/26: 2.9, 2.10 Introduction to Laplace Transforms, using them to solve IVPs.  Here is a Laplace Transform Table.

3/28: 2.11 Laplace transforms  of piecewise continuous functions.  Quiz 8 on 2.4, 2.5, 2.6, 2.8.

3/30: no class

4/2: 2.12  Laplace transforms  of impulse functions.  The Dirac delta function, which is not really a function but is a generalized function.

4/4: 2.13, 2.14, 2.15  Convolution, reducing systems to single diff eq, higher order.  Quiz 9 on laplace transforms.  Bring a copy of the table to use.

4/6: 3.1, 3.8, 3.10. Solving y'=Ay.

4/9: review 

4/11: Quiz on convolution, higer order linear diff eq, 2.13, 2.15

4/13: Exam #2 on chapters 1 and 2 of Braun. You may bring a 3x5 card with anything written on it, as well as your Laplace transform tables.  Here is a sample exam and solutions.

4/16: We looked at solutions to y'=Ay in the two dimensional case where A is diagonalizable with real eigenvalues.

4/18:  Quiz 11 on the material we covered 4/16.  We looked the remaining n=2 cases (complex eigenvalues, 0 determinent, nondiagnalizable). 3.9, 3.10, 4.7.

4/20: Introduction to behavior of autonomous systems y' = F(y).

4/23:  Example of autonomous system in R2 with 4 critical points.  Proof that for a two dimensional y'= F(y) at a critical point with complex eigenvalues a+-bi, with a and b not 0, the solutions are approximately logarithmic spirals.  Note that theorem 4 on 425 is weaker than it needs to be.  Astronger statement is here.
4/25: Matlab project announced.  Answered lots of questions. had quiz.  The first quiz problem was to draw solution curves around the critical points of x'=x2+y, y' = x+y.  Critical points have y=-x^2 so x-x^2=0 so x=0,1, y=0,-1.  The linearized system has matrix [2x 1; 1 1].  At (0,0) the characteristic polynomial is x^2-x-1 with eigenvalues (1 +- sqrt(5))/2 so there are positive and negative eigenvalues so it looks like a saddle.  At (1,-1) the char poly is x^2-3x+1 with eigenvalues (3+-sqrt(5))/2 so all eigenvalues are positive.  So the solutions look like those on the front cover of Braun with arrows pointing out.  the second problem was x'=3x-2y, y' = 2x-y.  The char poly is x^2-2x+1 so eigenvalue is just 1.  Eigenvectors are NS([2 -2; 2 -2]) so an eigenvector is (1;1).  So there are solution curves pointing out along the line x=y.  All other solutions are tangent to x=y as they approach the origin (with t approaching minus infinity) and look like figure 4b on page 422 with arrows reversed.  (note that if, say, x=0 and y>0 then solution curves point in the direction of (-2,-1) and so they go counterclockwise.

4/27:  Example of a pendulum with and without damping.

4/30: More on the pendulum.  Stability, assymptotic stability and unstability.

5/2: Quiz 13, similar to topics on 4/25 quiz, but also including stability.  Matlab progress report due. Point out that sometimes you can find a solution curve even if you cannot solve the differential equation.

5/4: review for exam 3.  Intro to Fourier series.

5/7: Exam 3. You can consult a 3x5 card.  the exam will be on sections 3.1, 3.8-3.10, 4.1-4.4, 4.7, and the fact that in the 2 dimensional case when no eigenvalue of the linearized system has 0 real part, the nonlinear and linearized systems behave similarly near the critical point (except that straight line solution curves may curve a bit).  Here is a sample exam with solutions.  Note we did not cover the material for problem 4, but read the solution anyway.

5/9: Final matlab projects due. Review.  Besides exam 3 material, here are some other problems and sections


 5/14:  I expect to be in my office 3111 much of the day if you have last minute questions.

 5/15:  Final exam 8-10 in 0302.  You may bring an 8.5"x11" sheet of paper with anything you want written on it, two sides. Also bring this Laplace transform table. Here is a sample final exam with solutions.