The animatrix is a collection of short stories with elements of the matrix. Quit artistic,
each has its own animation style. All based on stories by the Wachowski brothers.
This one is `Kid's Story' directed by Shinichiro Watanabe.
A school scene with some math content about linear difference equations. I just entered what I
saw on the board into google and found that the text has
been taken from these mathpages
of Kevin Brown who is one of the pioneers in math blogging.
The movie stops after "For example, the sequence tha...", but complete in the mathpages as
"It can be shown that this same summation applies to any (convergent) linear recurring
sequence with the initial values 0,0,...0,1,... (assuring
the characteristic polynomial has distinct roots). For example, the sequence that satisfies
the 3rd order relation s_{n} = 3 s_{n-1}-5s_{n}-2+7 s_{n-3}
consists of the values 0,0,1,3,4,4,13,47...
and the characteristic polynomial is f(x)=x^{3}-3x^{2}+5x-7. "
This text is a bit inappropriate for a general high school. The concept of solving general linear recursion
difference equations of arbitrary order is mostly done in college and also then not in this
forbidding way (but it is great for the story and similarly funny like the cubicle work in the
software company, in which Nemo worked). The math could work in high school like for the
very special linear recursion, the Fibonacci sequence F_{n} might appear. When I
was in high school, our Math teacher Roland Staerk
challenged the class to find a formula this famous sequence 1,1,2,3,5,8,13,21,... We tried but nobody
could find the Binet formula, a formula that can also be found without linear algebra:
with an Ansatz F_{n}=x^{n}, the Fibonacci relation F_{n+2}=F_{n+1}
+ F_{n} becomes x^{n+2}=x^{n+1} + x^{n}=0. The reason for the name characteristic
polynomial is that this produces the characteristic polynomial
x^{2}-x - 1=0, which has the golden ratio φ and its inverse 1/φ as solutions. The general solution
is therefore F_{n} = A φ^{n} + B/φ^{n} which is the Binet formula by fixing the constants
with n=0 and n=1.