**Re: MATH1231/1241/1251 SOS Thread**
Well I'm not sure why I used the word "screwed" any more because yeah that is a bit excessive. Still if they want to do really well I don't exactly think they should be asked on something they haven't been taught; you're not meant to assume the students will learn it themselves

If HSC maths questions (eg irrationality of e or pi) can involve proof by contradiction, why is it so obscene that questions from a first year uni course should? I am sure lots of high school students aren't formally taught proof by contradiction either, but this does not stop it from being in exams that are far more rigidly constrained by syllabus than any uni course.

In any case, they are not being "asked on" proof by contradiction, that is just the most convenient/natural way to phrase the argument. Another possible argument is as follows:

T: Span(S)->Span(R) is a linear map between vector spaces.

By rank-nullity:

dim(span(S))=dim(ker(T))+dim(im(T)) >= dim(im(T)) = dim(R)= 3 since the three vectors in R are linearly independent.

We also have dim(span(S)) =< 3, since it is the span of 3 vectors. Hence dim(span(S))=3 and so the spanning set {v1,v2,v3} must be linearly independent.

This is far less elegant than the previous argument, but it is still an easy proof that a first year could construct which is not a proof by contradiction.