So, this post is going to consist of an awesome thing my math lecturer said today. There are two versions of the introduction- one for people who have knowledge of Linear Algebra, and one for those who don't. (Incidentally, this is the lecturer that claims
he was "was almost as adorable [at the age of three
] as I am now
, hard as that may be to believe.")
Math Heavy Intro:
So we're learning about infinite dimensional vector spaces in linear algebra (as opposed to finite dimension vector spaces like lines and planes through the origin), and one example that came up in class was the vector space of all real functions defined on x=0 to x=1 such that f(0)=f(1)=0. We defined f(x).g(x) as the integral of f(x)g(x) over this interval, and then started talking about how the dot product is only zero if f(x) is zero, but then came up with a counterexample of f(x)=0 except for at a single point, where it was non-zero. At this point we decided to exclude functions with point discontinuities.
We're doing some math-y stuff in Linear Algebra. We defined some cool properties for equations that also exist for matrices, and were happy when they worked, but then found weird counter examples. They made us sad.
The Steven Johnson
Explanation of this sad fact:
So this is the problem with infinite-dimensional vector spaces. They work fine as long as you're reasonable and have well behaved functions, but you can always construct perverse counterexamples for which they don't work. These never come up in real life, but this is what mathematicians like to do, come up with these weird contrived examples.
Well, in this class we're not going to be perverse. We're going to follow the Google model: Don't be evil.
He is the greatest thing EVER.
Also, you should look at the links. They are filled with silly pictures of him.