Linear Algebra Class: What I've Learned So Far

[BoundByAxioms]

I am currently enrolled in a linear algebra class that started at the beginning of September, and it will end in about 4 weeks. We are using Shifrin and Adam's Linear Algebra: A Geometric Approach. The professor is great: he's a great teacher who really knows his stuff. However, it feels as though we are going laboriously slow though the class. Since September, we've only covered Chapter 1: Vectors and Matrices (Vectors, dot products, hyperplanes in Rⁿ, Systems of Linear Equations and Gaussian Elimination, The Theory of Linear Systems) Chapter 2: Matrix Algebra (Matrix operations, Inverse matrices, the transpose) and 4 sections of Chapter 3: Vector Spaces (Subspaces of Rⁿ, Linear independence, Basis and Dimension, and The Four Fundamental Subspaces - we haven't covered graphic examples or abstract vector spaces in the chapter). And we'll probably finish up with Chapter 4: Projections and Linear Transformations in a few weeks. We haven't (and very likely will not) cover the chapter on determinants, nor the chapter on eigenvalues and eigenvectors. Have I been exposed to enough linear algebra to have a decent grasp on the subject?

 

[WarPhalange]

Whoa!

Determinants aren't hard, but they are used often to check if a given equation has a solution, among other things.

Eigenvalues and Eigenvectors aren't hard either, but they are so important in quantum mechanics that it's a crime to omit them in a Linear Algebra class. Doing/using them is fairly easy, but it's the concept of them that should get ingrained into your brain because it is very powerful.

 

[BoundByAxioms]

I feel like I have a pretty strong theoretical foundation in linear algebra, but maybe too much for an undergraduate? It's a 300-level class.

 

[WarPhalange]

I took LA in community college and we covered Determinants and Eigenvectors/values.

However, it also depends on how the class is taught. My professor lectured to the board, so I don't remember how the class was taught. I just remember the problems in the book were "Do x" or "show y", and not really strict proofs. Community colleges are geared towards engineers more than physicists. Not that I like or care about proofs, but, if that's taking up a lot of your time, then you're not moving "slow"... how can I put this... okay, you're covering the same "area" of knowledge, except that yours is going to be a horizontal rectangle vs. my vertical rectangle. Know what I mean? I went further into it, but not as deep, essentially.

 

[BoundByAxioms]

I took LA in community college and we covered Determinants and Eigenvectors/values.

However, it also depends on how the class is taught. My professor lectured to the board, so I don't remember how the class was taught. I just remember the problems in the book were "Do x" or "show y", and not really strict proofs. Community colleges are geared towards engineers more than physicists. Not that I like or care about proofs, but, if that's taking up a lot of your time, then you're not moving "slow"... how can I put this... okay, you're covering the same "area" of knowledge, except that yours is going to be a horizontal rectangle vs. my vertical rectangle. Know what I mean? I went further into it, but not as deep, essentially.

Yeah I understand that analogy. Yeah, my college doesn't really offer an applied math type degree, just a pure math degree. On top of that my prof is a world class linear algebraist, and wants us to have a really thorough knowledge.

 

[quasar_4]

I'd hope to have a good grasp on vector spaces, linear mappings, inner product spaces, eigenvalues and determinants from an upper level linear algebra course.

If it's the required course that all the scientists and engineers take (i.e. not the upper level one), so long as you know how matrices, determinants, and eigenvalue problems work, you will be okay. Inner products are a big deal too, but you'll learn them in other classes as well. But if you're not solid on all of it, I'd recommend taking another LA course. It is SOOO wonderful to have a good grasp of linear, whether you're a physicist or engineer or whatever... it comes up everywhere. I know LA came up for me in probability, quantum mechanics, relativity, briefly in optics, my waves classes, etc., and it helped me a lot in abstract algebra, because it's all kind of the same game (find structures, study the structures, write proofs about them, etc.). It sounds like your class is doing fine, but I'd be sure to teach yourself eigenvalue problems - both very important and very useful.

 

[biggiekjac]

Haha I'm with WarP - eigenvalues and eigenvectors are pretty much the most important thing I took away from my linear algebra class. My professor for QM2 always made us repeat the mantra "The business of Quantum Mechanics is diagonalizing hamiltonians!" Hehehe