Linear Algebra: Understanding the Concept of Proof-Based Classes

[bor0000]

I am taking linear algebra right now, honours version. and i wonder if it is considered a real math course, i.e. is it really abstract or not, or is it like the baby stuff? i.e. the teacher gave in class a proof by induction that eigenvectors from distinct eigenvalues are independent. and so on. but this class is for physics majors, so i and many others are not taking analysis2 concurrently... i wonder if analysis 1&2 will be more abstract, i.e. harder, than this course, or the same? because i like this course, but I am afraid if it were much more abstract, i'd be in trouble. I am thinking of switching into the math major, then next fall i'll be taking analysis1 and algebra3, while most honours math students will be taking algebra3 and analysis3! but i think then i will also sign up for probability, which has analysis2 as a prerequisite.

 

[philosophking]

Linear Algebra at my school was about half what I think most professional mathematicians would call "computation" and half "rigor", i.e., about half of the math in the class is more calculus 1,2,3 kind of stuff, while the other half is more like what you see in an introductory real analysis course (at least, in the States).

Although, I've never heard of algebra or analysis 3, so I'm not quite sure what the equivalents are.

Analysis is pretty close to total "rigor" (little "computation"), for my classes. Most of my class is "Definition, Theorem, Proof" -style. There is just a continuous flow of theorems and proofs, whereas in linear algebra I remember a few more "formula, plug in" aspects.

 

[mathwonk]

as usual to find out what the standards are, READ A BOOK! The book by Hoffman and Kunze on Linear algebra is the standard for mathematicians linear algebra.

or download my book, from my webpage, http://www.math.uga.edu/~roy/, or the much better book by sharipov, http://www.geocities.com/r-sharipov/e4-b.htm.

I apologize there are errors in my book, and a newer version will be posted in a week or two.

 

[gravenewworld]

You will know when you take a proof based math class. In a proof based math class you will be given like 2 or 3 homework problems that will take you 4 hours to figure out. A proof based math class is where real math begins, it isn't simply a take the derivative and plug in 3 for x or take the integral of this type of math class. You will be required to think abstractly. If you want a taste of what real analysis is, open up your calc book. If it is any good you will be able to find an epsilon-delta proof for a limit, which isn't covered in a normal calc 1-3 class. Sure you calculated limits in calc 1, but you didn't prove what the limit was. Modern algebra aka abstract algebra is also a proof based math class.

 

[bor0000]

thanks! i looked at the webpage, but i don't really have time to read a book:( I've noticed from experience that for example looking at course notes or even final exams from prior years doesn't tell me anything, because if i don't have the knowledge to understand the questions, i can't even tell if a question is hard or easy.

actually in my class the teacher gave a cheap linear algebra book, but told us to rely more on the notes written in class. the link covers a lot of what we have covered, except it may be in a different order. i.e. we have not yet covered the eigenvectors of symetric matrices on page 8. but on page 10, i believe ci do not have to be distinct, as long as their geometric multiplicity=algebraic multiplicity. and we did prove in class(but I am a little behind, just doing homework on that topic right now) that those eigenvectors must all be independent. also we don't really use words like isomorphism and surjections. the clue words that i looked up for the description of algebra 3 are 'sylow theorem', and for analysis 1 (mean value theorem), analysis 2(riemann integration and sequences/series), analysis 3(multivariable calc and intro to metric spaces). most math majors here take analysis 1&2 and algebra1&2 during sophomore year. but I am not decided in my major, so instead I am taking algebra for physicists this spring. but if i do enroll into math next fall, i'll be taking also the probability course which has analysis 1&2 as prerequisites.

i can't do my math homework again:((( it is here
http://www.math.mcgill.ca/schmidt/247w05/ass4.pdf
i did the first part, by saying that A represents the coordinates of the transformation with respect to the regular basis for u and v. and then it's equal to D^-1*[T]bu,bv*C where C and D are the matrices of u and v with respect to the regular basis. but for the bonus part, i am stuck and confused. i guess it must be because i did not understand the concept as was supposed to!