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.[sighs]

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

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 A*

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 A

^{20}would only take 5 multiplications, since all one would need are the products A^{2}= AA, A^{4}= (A^{})^{2}, A^{8}= (A^{4})^{2}, A^{16}= (A^{8})^{2}, and A^{20}= A^{4}A^{16}.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 A

^{23}is more motivating, since 23 is prime.**q-:**

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