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 ..

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[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-: