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It seems like college Linear Algebra expeiences fall into one of two buckets: either you took a "Linear Algebra / Differential Equations" class where you learne

It seems like college Linear Algebra expeiences fall into one of two buckets: either you took a "Linear Algebra / Differential Equations" class where you learned no theory and saw a bunch of practical examples and did problem sets that were routinely north of 10 pages, OR you took a highly theoretical course where everything was far too abstract for your first year of college level of comprehension. I took the first route and retained none of the information, despite doing well in the course. Then I took computer graphics and cried all the way home. This link seems like a happy middle ground.

j2kun 12y ago
It's strange how many of my CS friends were deathly afraid of linear algebra, yet still enjoying their graphics classes.
- otoburb 12y ago
  What are your thoughts on those revisiting linear algebra to instead try their hand at learning geometrical algebra instead?
  - dfan 12y ago
    You can't really get around linear algebra. Geometric algebra is really neat, and you can benefit from taking a geometric algebra approach to linear algebra (Macdonald's book is pretty good for that), but it enhances linear algebra, it's not a substitute for it.
- jlarocco 12y ago
  I think I'm in that camp. I got a D in linear algebra, my lowest grade in university and one of my lowest grades ever, but excelled in computer graphics, and even got permission to take a graduate level course in it for elective credit as an undergrad. My recollection is that the problem sets in the linear algebra class seemed really abstract and vague, and I couldn't reason through it very well because I couldn't visualize what we learned, given the information from the class. Computer graphics, however, let me visualize what we were covered, and suddenly everything made sense. I could see, literally, what the end result was supposed to be, and it just clicked. Now that I think about it, it's kind of a shame I had the linear algebra class first. The CG experience would have helped me out quite a bit.
nextos 12y ago
I found Linear Algebra Done Right by Axler to be a fantastic theoretical text that makes everything click together from day 1. It's derived from his paper Down with Determinants. He starts by explaining that linear algebra is about studying the properties of linear transformations over vector spaces, and then goes on to explain what this means. In many theoretical books it takes ages to get this message, and in some practical ones you get lost in a sea of matrices without even understanding what a matrix actually is. Furthermore all his proofs are very slick.
- jjoonathan 12y ago
  Seconded. I was very skeptical of LADR when I was forced to use it freshman year. I thought I already knew all the practical stuff from Strang's book, but learning linear algebra from the mathy angle has paid off again and again and again. Choices of units & coordinate systems are impossibly confusing without having a foundation in how to reason independently of such arbitrary choices and add them in later. I think the turning point came when I realized that "abstract" linear algebra was actually much closer to physical reality than "applied" linear algebra. Superficially, it seems like the exact opposite state of affairs from computer engineering, where grids of numbers take you closer to "bare metal" and "higher level reasoning" takes you further away. In math, it's the grids of numbers that are further removed from "bare metal" and the abstract reasoning that you can count on when all else fails! I didn't need to be convinced that I would eventually come to regret having an insecure foundation so much as I was mixed up about up vs down and trying to build a foundation in the middle of the sky.
- mafribe 12y ago
  In my experience, Linear Algebra Done Right by Axler is a great second book on linear algebra. But it makes sense to most students only if they've seen and done concrete matrix calculations before. Giving it to a mathematical novice is likely to lead to baffelment. Too abstract.
  - nextos 12y ago
    I don't think it's too abstract. It contains plenty of geometrical intuition. And it makes things very clear and he says them very _early_. E.g., a matrix represents a linear transformation. It took me ages to understand this at high school because I was lost in a sea of matrices.
deleted 12y ago
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j2kun 12y ago

It's strange how many of my CS friends were deathly afraid of linear algebra, yet still enjoying their graphics classes.

otoburb 12y ago
What are your thoughts on those revisiting linear algebra to instead try their hand at learning geometrical algebra instead?
- dfan 12y ago
  You can't really get around linear algebra. Geometric algebra is really neat, and you can benefit from taking a geometric algebra approach to linear algebra (Macdonald's book is pretty good for that), but it enhances linear algebra, it's not a substitute for it.
jlarocco 12y ago
I think I'm in that camp. I got a D in linear algebra, my lowest grade in university and one of my lowest grades ever, but excelled in computer graphics, and even got permission to take a graduate level course in it for elective credit as an undergrad. My recollection is that the problem sets in the linear algebra class seemed really abstract and vague, and I couldn't reason through it very well because I couldn't visualize what we learned, given the information from the class. Computer graphics, however, let me visualize what we were covered, and suddenly everything made sense. I could see, literally, what the end result was supposed to be, and it just clicked. Now that I think about it, it's kind of a shame I had the linear algebra class first. The CG experience would have helped me out quite a bit.

nextos 12y ago

I found Linear Algebra Done Right by Axler to be a fantastic theoretical text that makes everything click together from day 1. It's derived from his paper Down with Determinants. He starts by explaining that linear algebra is about studying the properties of linear transformations over vector spaces, and then goes on to explain what this means. In many theoretical books it takes ages to get this message, and in some practical ones you get lost in a sea of matrices without even understanding what a matrix actually is. Furthermore all his proofs are very slick.

jjoonathan 12y ago
Seconded. I was very skeptical of LADR when I was forced to use it freshman year. I thought I already knew all the practical stuff from Strang's book, but learning linear algebra from the mathy angle has paid off again and again and again. Choices of units & coordinate systems are impossibly confusing without having a foundation in how to reason independently of such arbitrary choices and add them in later. I think the turning point came when I realized that "abstract" linear algebra was actually much closer to physical reality than "applied" linear algebra. Superficially, it seems like the exact opposite state of affairs from computer engineering, where grids of numbers take you closer to "bare metal" and "higher level reasoning" takes you further away. In math, it's the grids of numbers that are further removed from "bare metal" and the abstract reasoning that you can count on when all else fails! I didn't need to be convinced that I would eventually come to regret having an insecure foundation so much as I was mixed up about up vs down and trying to build a foundation in the middle of the sky.
mafribe 12y ago
In my experience, Linear Algebra Done Right by Axler is a great second book on linear algebra. But it makes sense to most students only if they've seen and done concrete matrix calculations before. Giving it to a mathematical novice is likely to lead to baffelment. Too abstract.
- nextos 12y ago
  I don't think it's too abstract. It contains plenty of geometrical intuition. And it makes things very clear and he says them very _early_. E.g., a matrix represents a linear transformation. It took me ages to understand this at high school because I was lost in a sea of matrices.

deleted 12y ago

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