Tuesday, April 05, 2011

they weren't kicking and screaming, but ..

in my lectures yesterday i was too technical, having dragged my students through some gory computations:

in linear algebra, computing the 20th power of a 3 x 3 matrix, via diagonalιsation, took 20+ minutes.
(admittedly, i started wondering whether it would be faster just to have multiplied the matrix 20 times .. [1])

as for the proofs class, we spent essentially the whole lecturing proving that every positive number has a square root.
the proof involved two lemmas on the fly. at the time, i thought it would be rather artificial to prove them in advance. instead, i had hoped that by proving those results as they were needed, the students would get a better sense of problem-solving strategies.

by the end, i couldn't help but suspect that my students just wanted it to be over .. \-:
[sighs]

when i was an undergrad, i remember learning theorems that took a whole week of lectures to prove ..

[1] for a square matrix A, computing A20 would only take 5 multiplications, since all one would need are the products A2 = AA, A4 = (A)2, A8 = (A4)2, A16 = (A8)2, and A20 = A4A16.

now that i think about it, is 5 the minimal number of multiplications? combinatorιcs isn't my strong point, and right now i'd rather go back to working on a research question regarding rectifiabilιty of sets.

at any rate, perhaps A23 is more motivating, since 23 is prime.
q-: