It’s so nice to see mathematics playing an important part in popular television. In tonight’s Dr Who the spaceship crew had to find the next number in the sequence 313, 331, 367, …. Dr Who recognises this as a sequence of happy primesÂ with the next one being 379. It’s all explained at that Wikipedia articleÂ and a longer version of the sequence can be found at the wonderful The On-Line Encyclopedia of Integer Sequences.
It’s interesting that the Dr Who reference was put in the Wikipedia article half-an-hour before the programme was aired so probably an inside job. I was delighted thatÂ in the programme Dr WhoÂ asked if mathematics was so dumbed down that recreational maths wasn’t studied any more. As Russell T.Â Davies, the head writer,Â said in tonight’sÂ Dr Who Confidential, the programme reflects current concerns, so this problem has clearly reached a wider audience than I imagined was the case.
Most (all?) undergraduate courses use determinants to introduce eigenvalues and eigenvectors. So the eigenvalues of a matrix (or linear transformation ) are the solutions of . However, Sheldon Axler published a paper in 1994 called Down with Determinants! where he maintains that determinants should not be used so early in linear algebra courses. He gives a very nice proof of the existence of eigenvalues in finite-dimensional vector spaces (over ) which I would like to reproduce here.
Every linear transformation of a finite-dimensional complex vector space has an eigenvalue.
Here is his proof:
Let be a non-trivial finite-dimensional complex vector space and tÂ a linear transformation . Let be a fixed non-zero vector in and suppose that . Then the vectors are linearly dependent. Hence there exists complex numbers not all 0 such that
Now, since is algebraically closed, Â the polynomial will factorise so we get
where are complex numbers with . It follows that
which means that, since this is composition of functions, and , then
soÂ Â Â so that and is an eigenvector
so Â is an eigenvector
is an eigenvector,
and hence has an eigenvalue. Â Â
The Art of Problem Solving (AopS) site encourages, teaches and promotes mathematics competitions from small local ones right up to the International Mathematical Olympiad (IMO). As it says on its front page:
Is math class too easy for you? Looking for a greater challenge?
You’ve come to the right place.
It has an excellent introduction to siteÂ and a forum to discuss problems. The forum has an RSS feed where students pose new problems every day. So if you’re “looking for a greater challenge” subscribe to this feed. Here is a random sample of some of the problems posed recently, which range from very easy to extremely difficult. Click on the problem number to go to the discussion on it.
- 1. We define addition in a different wayÂ to usual; an addition statement is true only if the letters in the addends is a rearrangement of the letters in the sum. For example,
10 + 6 = 16?
TEN + SIX = TENSIX = SIXTEN, but to be 16 it would need another E.
Find a “true” addition a + b = c + d.
2. Prove that .
3. Let be nonzero real numbers. Find all ordered pairs such that .
4. is a continuous complex-valued function satisfying:
5. If and are relatively coprime, find all possible values of .
6. Let be three angles ofÂ . Prove that .
7. For each function which is defined for all real numbers and satisfies and determine the value of .
8. Let be positive reals such that . Prove that .
9. For , we define the numbers . Find the last digit of the number .
10. The product of several distinct positive integers is divisible by . Determine the minimum value the sum of such numbers can take.