Linear algebra courses at MIT from Prof. Gilbert Strang

Linear algebra is fundamental mathematical knowledge for those who need to perform computational natural sciences. It is a neat formalism to express things in a compact way, and describe precious algorithms to solve computational problems from chemistry, physics, astronomy, and so on.

I found these precious and very clear lectures from MIT professor Gilbert Strang. The lectures are available to the general public under the MIT OpenCourseWare distant learning initiative. I am pretty sure not to say anything excessive when I claim that such initiatives should be declared Intangible Heritage of Humanity due to their important role in scientific advancement and education.

Here is the list of the lectures, linked to the proper material on the MIT OCW website:

  1. The geometry of linear equations
  2. Elimination with matrices
  3. Multiplication and inverse matrices
  4. Factorization into A=LU
  5. Transposes, permutation, spaces R^n
  6. Column space and nullspace
  7. Solving Ax=0: pivot variables, special solutions
  8. Solving Ax=b: row reduced form R
  9. Independence, basis and dimension
  10. The four fundamental subspaces
  11. Matrix spaces; rank 1; small world graphs
  12. Graphs, networks, incidence matrices
  13. Quiz 1 review
  14. Orthogonal vectors and subspaces
  15. Projections onto subspaces
  16. Projection matrices and least squares
  17. Orthogonal matrices and Gram-Schmidt
  18. Properties of determinants
  19. Determinant formulas and cofactors
  20. Cramer’s rule, inverse matrix, and volume
  21. Eigenvalues and Eigenvectors
  22. Diagonalization and powers of A
  23. Differential equations and exp(At)
  24. Markov matrices; Fourier seriesQuiz 2 review
  25. Symmetric matrices and positive definiteness
  26. Complex matrices; fast Fourier transform
  27. Positive definite matrices and minima
  28. Similar matrices and Jordan form
  29. Singular value decomposition
  30. Linear transformations and their matrices
  31. Change of basis; image compression
  32. Quiz 3 review
  33. Left and right inverses; pseudoinverse
  34. Final course review
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XML Namespace URL (updated)

I am attending the EMBRACE workshop in SOAP clients. A very interesting workshop I would say, and I’ll wrote more about it when finished. During the workshop, it has been pointed out the usual issue of XML namespaces: the attribute value looks like a URL but it is not referring to anything in particular. An example from a SOAP envelope

<?xml version=”1.0″?>
<soap:Envelope xmlns:soap=”http://www.w3.org/2001/12/soap-envelope” soap:encodingStyle=”http://www.w3.org/2001/12/soap-encoding”>
</soap:Envelope>

Technically, the attribute value is a string that represents the namespace. The choice to use a URL stems by the fact that the DNS system guarantees uniqueness and authoritativeness, so if you define a new namespace, using your own domain guarantees (sort of) to be unique. However, there is no consensus about what this URL should resolve to. In some cases, it refers to the XML schema. In others to a DTD, or to a stylesheet, or more frequently to nothing. You can have a picture of the situation from these articles 1, 2, and 3. The first two articles, in particular, advocate the use of RDDL to solve the problem. Basically in the ambiguity of what to put, the answer with RDDL is: none of them. Instead, provide a RDDL document that says where to find each of them (if provided). Not a bad idea.

Waiting for the community to decide what to put at that address, my personal choice went toward a still standard but not deliberately confusing choice. I put a uuid URN.

xmlns:foo=”urn:uuid:212e2ac7-dc35-4112-ae86-cefd26abb856″

which is valid according to standard (you can put any URN), it is unique and at least it does not pretend to look like it’s referencing to something. You can generate one with uuidgen.

An objection to this approach is that a uuid is not easy to remember, and so you have to copy and paste it every time. Well, the URL approach has more or less the same issue. Namespacing does not work correctly if you don’t specify the URL namespace exactly, so you end up copying and pasting it anyway.

Update: Just today, the W3C released an interesting plea to developers in order to limit the traffic at w3.org. Apparently, they get an insane amount of traffic, due to the attempts by various softwares around the net to get the documents referred by the addresses in DTD and namespace. This is another indication that you should be very careful in putting an URL, in particular if your format becomes very popular and you don’t have big pipes to hold the traffic.

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Course started in bioinformatics

I just started to hold exercises lectures during a course in bioinformatics. I am publishing the lecture notes while the course goes on. Please note that the copyright holder is DTU (although my preference goes to a CreativeCommons license), and that there could be mistakes on the notes, mainly because I am preparing them in a big rush.

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