Linear algebra nugget of the day. You can classify a 2d quadratic form without finding eigenvalues. Det tells you if the eigenvalues are the same sign and if det is positive, trace tells you which sign. Exercise: what if det is 0?
Alas, it is now common for a first course in linear algebra to omit (or at least to downplay) such subtleties as this. One is likely to see them stressed only in a second course -- and even there, one cannot be sure to see them at all. (Linear algebra is actually quite hard.)
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I took a linear algebra course, but I don't remember this. I remember doing operations on matrices and work with summation notation. I got stuck on sums of infinite series, & after taking philosophy realized I was thinking like Zeno (as in the paradoxes). Always more to learn!
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