# paul

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 Feb25 awarded Nice Question Feb2 awarded Notable Question Oct31 revised Graduate school self-doubt (currently an undergraduate) removed details Oct31 awarded Citizen Patrol Jun14 awarded Yearling Mar27 awarded Popular Question Feb28 awarded Nice Question Jul28 accepted Why are $\sin$ and $\cos$ (and perhaps $\tan$) “more important” than their reciprocals? Jul25 awarded Announcer Jul25 comment Why are $\sin$ and $\cos$ (and perhaps $\tan$) “more important” than their reciprocals? @JonasMeyer Perhaps. I've always thought that we prefer rise/run because European languages go from left-to-right then top-to-bottom, which in turn made the horizontal [respectively, vertical] axis represent the independent [respectively, dependent] variable, no? In which case, it is still, in an admittedly forceful way of pushing it, arbitrary. Jul25 comment Why are $\sin$ and $\cos$ (and perhaps $\tan$) “more important” than their reciprocals? @Mike Indeed! My next question is then: coincidence or design? Jul25 asked Why are $\sin$ and $\cos$ (and perhaps $\tan$) “more important” than their reciprocals? Jul19 accepted What makes elementary row operations “special”? Jul19 comment What makes elementary row operations “special”? ;) I discovered this a while ago, too, only to have my high school math teacher tell me I was wasting time "using less tools when I had more." Jul19 comment What makes elementary row operations “special”? silly me for thinking it's finitely generated Jul19 comment What makes elementary row operations “special”? Can it be (easily) proven that three is the fewest number of generators needed to obtain the entire group of invertible square matrices? Edit: It's obviously not cyclic, because multiplication is not commutative, so can we come up with two (probably more complicated) matrices that generate the three "canonical" elementary matrices? If not, why not? (I'm unfamiliar with the etiquette here, does this question scream "ask new question"?) Jul19 comment What makes elementary row operations “special”? Thank you, this is the answer I needed. So in some sense, they're the "most natural" of all arbitrary generators one can come up with? Jul19 comment What makes elementary row operations “special”? Just to be clear, are you saying that all linear systems with the same solution space can be obtained from one another after a finite number of elementary row operations? Or am I misunderstanding something here? Jul19 comment What makes elementary row operations “special”? Yes. "Preserving the solution space" is a nice property they possess. But are these the only three operations that have this property? For one thing, we can compose them to get other such operations. My question is, what makes them so elementary? Jul19 asked What makes elementary row operations “special”?