Longest. Unit. Ever. Or at least it feels like it, because I’ve been working through Unit LA3 on and off for about a month now! I was out of action with a cold for a couple of weeks in the middle, and when I came back to it I found that whatever understanding of vector spaces I had in the first place had completely disappeared. Perhaps I sneezed out the relevant brain cells…
I think the thing I had trouble with the most was spanning sets – particularly proving that a particular set spans a given vector space. I found the Vector Spaces chapter of Paul Dawkins’ excellent Linear Algebra class notes really helpful (though of course you’ve got to beware of the different notation used – I don’t want to incur the wrath of my tutor by using non-M208 notation!).
Linear independence took a bit to sink in, but I think I’ve just about got the hang of it, and bizarrely enough I found the later sections about orthogonal and orthonormal bases much easier than the earlier stuff. I’m quite intrigued by the remark at the end of the unit about orthogonality and polynomials – the idea of two polynomials being orthogonal to each other is very hard for me to get my head around, but I’d be interested in doing some more work on it. I wonder if it will turn up in any of the level 3 pure maths courses, or whether it’s more of an “applied” topic. The unit also mentioned that orthogonal polynomials are important in mathematical physics, so perhaps Alex will end up encountering them in one of his future courses.
In other news, I got my marked TMA01 Part 1 back last week, which came in at a nice 91%. I just hope I can manage the same kind of mark for Part 2. I didn’t get a great deal of feedback on Part 1 (which is understandable, since it’s only two questions), but my tutor did make a vague suggestion that I would be better off not word-processing my assignments in future. Unfortunately for her, I’ve already got TMA02 word-processed and printed out, ready to go! To be honest, I much prefer word-processing maths assignments – I hand-wrote all my TMAs for MST121, and had to scan or photocopy each one if I wanted a back-up copy in case the original got lost in the post. Very tedious indeed! So I’m not keen to go back to hand-writing assignments any time soon – they’ll have to pry my copy of MathType Lite from my cold, dead hands!
Just come across your site. It’s interesting to read for me as an M208 tutor because it allows me to view the course through the eyes of an enthusiastic student.
I sympathise with your tutor’s comments about word processing TMAs but there is a better way which still allows you to use a computer but without word processing. Instead you can typeset it to produce a professional quality document by using LaTeX. You are already using WordPress’s LaTeX add-on so how about going one step further and writing whole documents with it?
I must admit there’s a steep learning curve so it needs time and I definitely wouldn’t want it to disrupt your maths studies. Do visit OUSA’s LaTeX and TeX forum where, amongst other things, you’ll find a LaTeX TMA template that students have produced.
Thanks Steve, I’ll have a look at the LaTex forum. I’m interested in learning LaTex eventually, probably in the break between M208 and next year’s presentation of M381. Definitely seems like the way to go, if I’m going to make a habit of doing maths courses!
The orthogonality of polynomials and Trignometric Functions is a key concept in Applied mathematics. The simplest example is sin (mx) and sin(nx) when you take the integral of the product of these functions from -pi to + pi is pi when m = n and 0 other wise. Normalising the functions by multiplying the sin function by 1/sqrt(pi) means that the functions can be seen as basis vectors for an abstract vector space. The equivalent of the scalar product being the Integral. This has profound implications utilised by sound engineers (amongst many) in that any complex waveform can be reduced to a sum over the fundamental basis vectors. Such a decomposition is called Fourier Analysis. If you do MST209 you will get an introduction to this.
However it turns out that there are a whole class of functions associated with the series solution to differential equations which can be treated in a similar manner. The names of the two most famous being Bessel Functions which usually arise in conjunction with problems in cylindrical coordinates (the transmission of light down optical fibres being a classic example) and Legendre Polynomials which arise when problems occur in spherical coordinate systems such as the Angular momentum states of the hydrogen atom in quantum mechanics.
The general study of this analogy between orthogonal polynomials and ordinary vectors is called Functional Analysis. In the Old days the OU did a course M201 which went into this in much more detail than M208 or MST209 does. Unfortunately its no longer avaiable.
PS Only just noticed this post so sorry for the delay
Best wishes Chris and good luck with the end of M208