Comment by jules
6 years ago
I like this idea. It's also how the subject developed historically. That said, don't you get a very computational book in which students spend a lot of time working with matrices filled with numbers if you teach Gaussian elimination first? How do you make sure that students understand the big picture? After all, modern linear algebra is a lot more than the Gaussian elimination algorithm. I'd even go so far as to say that if a student don't understand Gaussian elimination but understands everything else and that there exists some algorithm to solve linear equations, they'd still get 95% of the value of a linear algebra course.
Maybe you can do that if you make it very visual like the book linked in this post, so that you bring in the geometric concept of a vector and link it to linear equations, rather than only working with tuples of numbers. For example, linking the visual concept of linear independence of a set of vectors to a system of equations having a unique solution. If you do that you may be able to teach linear algebra from a linear equation solving point of view, while not getting bogged down in manipulating matrices of numbers.
> After all, modern linear algebra is a lot more than the Gaussian elimination algorithm
This is kind of my point - there actually isn't a whole lot more than Gaussian elimination at the core of linear algebra until you get to rings and modules. Of course I don't mean that in the sense that there aren't important definitions or topics. The theory is very rich. What I mean is that most theoretical questions you can ask about vector spaces and linear maps (or more abstractly, the linearity of a thing), are reducible to a matrix representation and Gaussian elimination.
You don't have to use Gaussian elimination, and I'm not arguing the course should become rote computation of matrices. There is a middle ground. For example if I hand you a linear map and ask you about the dimension of its range, you can identify a spanning set, refine that to a basis and you have your answer. You could also just take the rank of the matrix representing that linear map reduced to row echelon form. I don't think you lose much theory one way or the other.
But I could be wrong - if you have a specific example I'd be interested in hearing it. I've tried to declare front and center that I'm biased from learning linear algebra through Hoffman & Kunze. Halmos also grounded his (very theoretical) book in a generous use of matrices. Not covering matrices until later is a fairly recent development, and I think it's because (for reasons I've stated elsewhere) many professors don't distinguish between the rote computation of matrices and the rich theory of matrices.
Yeah, I agree that most questions in introductory linear algebra are reducible to Gaussian elimination. However, if I had to play devil's advocate, I'd say:
1) Gaussian elimination is only one of the algorithms for solving linear equations. Why that one? Perhaps there is a better one for pedagogical reasons.
2) Why focus on the algorithm in paricular? Maybe we could formulate the thing Gaussian elimination is doing more abstractly with less reference to a particular algorithm. For instance: any matrix A can be written as LDU where L is strictly lower triangular, D is diagonal, U is strictly upper triangular. Or maybe: any linear map can be written as a projection on the first k coordinates relative to some basis, where k is the rank. Or maybe there is a geometric way to understand it.
3) Why focus on this particular concept in particular? That concept is just one of many concepts in introductory linear algebra. Although you can reduce everything in introductory linear algebra to it, you could also pick some other concept around which you could center the course, such as linear independence, the determinant, or something else. Or some other decomposition, such as A = XDY where X,Y are products of elementary row operations (so they are square and have determinant 1), and D is a rectangular diagonal matrix. This decomposition is arguably more important than the LDU decomposition associated to Gaussian elimination. Why even focus on one concept in particular?
4) This is only for the very basics. You also need a plan for Gram-Schmidt, eigenvalues, spectral theorem, Jordan decomposition, and so on, which form the meat the linear algebra courses.