email@example.com (Dave Rusin) wrote in message news:<firstname.lastname@example.org>... > In article <email@example.com>, > Anonymous wrote: > >I'm currently a math major and am taking linear algebra, but I'm in > >serious danger of failing. I just don't get it! Is this newsgroup a > >place to come to ask questions and get information about learning > >math? Or is there somewhere more appropriate to go? I've always had > >trouble with vectors, and I think I fell apart sort of right at the > >beginning of linear algebra (although, I did manage to get a B- on the > >very first exam). I've got another exam next week. What can I do? I > >don't get all the terms, concepts, and jargon. Anyone know how to > >make learning linear algebra easier and more practical? Anyone got > >any practice problems? > > I would bet that the single practice problem you need to work on is, > "What is a vector space?" > > Our students "do well on the very first exam" because that's the part > of the course where we warm up with techniques for solving linear > systems of equations and such topics. But our LA course is also our > students' first course in which abstractions, axioms, and proofs play > a significant role. They often stumble because (among other problems) > they don't realize they need to _memorize_ definitions _precisely_. > So you can do a little self-assessment here to figure out whether > what you're missing is bits of topics or the core idea: can you, > right this minute, define what a vector space is? > > dave
Dave, this may be my problem. I did decent on the first exam because, like you said, it was solving linear systems, echelon form, linear dependences--easier stuff like that. I've always been a little confused with the concepts, but I think I may finally be getting a grip on what a vector space is:
It's a group of vectors that can be multiplied by any scalar and/or added together in any way, and whatever possible combinations that can result is the "vector space" for that group of vectors. This is how I understand it. For vectors in R^2, a plane is formed ("spanned"???) by the vector space. For vectors in R^3, a solid area is formed by the vector space. It gets difficult for me to move into dimension 4. While I understand that the same concepts hold, there's no more physical picture I can use to visualize what's happening. Is my understanding of "vector space" sufficient enough? Am I missing anything?
I know I am still struggling with the concepts of "span" and "basis". The weird thing is that I'm alright with the more advanced stuff; matrices, determinants, eigenvalues, eigenvectors. I'm a little hazy with diagonalization because it's the newest thing we've done. I know it's got something to do with the eigenvalues of a special type of matrix.
I guess I could also really use some help with understanding how a mapping gets converted into a matrix, and then how to solve it. I understand matrix multiplication and can do it well. But the concepts of image, kernel, and isomorphism and how they relate to the mappings/matrices seem to be lost on me. The odd thing is that I fully understand the definitions of "kernel" and "image" as they were applied in algebraic structures, but I don't get how they apply to linear really. "Isomorphism" is a concept I never understood in algebraic structures or linear algebra.
Thanks to the people who posted book suggestions, but I'm hesitant about buying any other books. I already bought the text, the Cliff's Notes guide to linear algebra, a 2003-version of the Schaum's outline, and I even have an old 1968 version of Schaum's that my grandmother used when she majored in math. Cliff's has been helpful, but too basic. Schaum's seems almost too advanced; it's great that they solve all the problems, but sometimes the explanations are lacking. I find that I do much better at math problems if I can first figure out how to solve a certain type of problem and then go back and try to understand the concepts behind it, rather than the other way around. Schaum's examples don't allow for this, because they assume you've already read (and understood) the concepts behind how to solve certain problems.
Maybe, if it isn't too much to ask, would anyone here be willing to post some problems relating to mappings/kernel/image/isomorphims and/or eigenvalues/eigenvectors, and I can attempt to solve them with your help?