Comments on: Vector spaces

By: Chris Finlay

Chris Finlay — Mon, 30 Aug 2010 19:17:38 +0000

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

By: Rob

Rob — Sat, 10 Apr 2010 14:13:05 +0000

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!

By: steve

steve — Sat, 10 Apr 2010 10:15:43 +0000

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.