Answer by Gareth for Why linear algebra is fun!(or ?)

2010-07-30T22:55:07Z

I like using the example of magic squares when starting to go over linear algebra, usually starting with $3\times 3$ squares. They're a nice recreational maths thing that everyone has seen before, but usually not thought about.

When asked for an example, most students come up with something like $\pmatrix{6&1&8\cr 7&5&3\cr 2&9&4}$, remembering a construction from before. When prodded for a second example, someone might suggest rotating or reflecting this example. Once it's suggest that we just want the rows, columns and diagonals to sum to the same thing, and that the numbers don't have to be distinct, someone usually thinks of $\pmatrix{1&1&1\cr 1&1&1\cr 1&1&1}$.

It then usually becomes clear that linear combinations of what we have so far will also work, and this leads naturally into asking how many squares we need in a basis, and so on. (I then ask them to work out the dimension of the space of $n\times n$ magic squares as homework.)

Another "unexpected" use of linear algebra is when they're asked to prove that things like $\sqrt2+\sqrt3$ or $\sqrt2 + \sqrt[3]2$ are algebraic. Many fiddle around until they chance upon an arrangement that works, but they all like it when we show that it's sufficient to take a few powers and say "oh, some combination of those will do". This usually goes down well, as people often like playing with numbers.

Answer by Gareth for Function with range equal to whole reals on every open set

2010-07-20T12:49:18Z

Once we have the idea of prodding the decimal expansion, there are any number of things we can do. Here are a few. We may assume we're starting with $0.a_1a_2a_3...$, e.g. by taking the fractional part of the input. If any of the sums/limits mentioned below fails to exist, map the input to $0$.

We construct the decimal digits of the image $b_0.b_1b_2b_3...$ one at a time. Consider the alternating sum of the $a_i$ for $i$ odd. If this converges (i.e. these $a_i$ are $0$ from some point on), then define $b_0$ to be this. Next, look at the alternating sum of the $a_i$ with $i\equiv 2 \mod 4$. If this converges to a number in ${0,...,9}$, then define $b_1$ to be this. Then construct $b_2$ using the $a_i$ with $i\equiv 4 \mod 8$, and so on.
Map the number to $\sum \frac{(-1)^{a_n}}{n}$.
Map the number to $\lim_{n\to\infty} (a_1+...+a_n)/n$. This will give us results only in $[0,9]$, but that's sufficient.

In each case, whatever tiny interval we're forced to start in, we can always get any output we like by using later digits.

User gareth - MathOverflow

Answer by Gareth for Why linear algebra is fun!(or ?)

Answer by Gareth for Function with range equal to whole reals on every open set