My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.
I was also taught linear algebra this way, by an applied mathematician with a background in chemical engineering:
- start by solving Ax=b with row reduction
- develop theorems about linear independence and spanning sets of vectors based on these exercises
- introduce the determinant from the perspective of linear systems (rather than eg geometry or group theory)
- eigenvectors and eigenvalues
Later I switched from physics to math and TAed a more “algebraic” approach involving groups/rings/fields. But the matrix-first approach was more helpful for both my physics coursework and later courses in numerical linear algebra.
I took like 3-4 courses in the US involving the engineering approach, starting in high school and continuing through the college as a CS major. That was all that was required.
But I also like algebra, so I happened to take a 400-level course that only math majors take my senior of college. And then I got the group theory / vector space view on it. I don't think 95% of CS majors got that.
I don't think one is better than the other, but they should have tried to balance it out more. It helps to understand both viewpoints. (If you haven't seen the latter, then picture a 300-page text on linear algebra that doesn't mention matrices at all. It's all linear transformations and spaces.)
What country were you taught in? Wild guess: France?
A book I enjoyed is Axler's Linear Algebra Done Right[0], in which, if I remember correctly, doesn't contain a single matrix.
[0]https://zhangyk8.github.io/teaching/file_spring2018/linear_a...
It does have plenty of matrices. The main thing it really does is avoid determinants until the very end. The determinant is certainly something I remember learning as a kind of rote operation, without really understanding any intuition behind why you'd multiply and add these numbers in this particular way. I still feel lacking in "feel" here, which is why I suppose I'm going through Axler now.
For example, I remember looking at the linear algebra book my department had used previously. Early on, it introduced the concept of the transpose of a matrix:
https://en.wikipedia.org/wiki/Transpose
Superficially, it looks like something good to introduce. It is fodder for easy homework exercises, and there is a satisfyingly long list of formal properties satisfied.
But why? What does the transpose mean? For what sort of problem would you want to compute it?
There are good answers to these questions (see the "transpose of a linear map" section of the Wikipedia article I linked), but they are not easy for a beginner to the subject to appreciate.
> You are probably about to begin your second exposure to linear algebra. Unlike your first brush with the subject, which probably emphasized Euclidean spaces and matrices, this encounter will focus on abstract vector spaces and linear maps.
It's not universal.
The US is a very big place. I doubt there is an american approach to linear algebra. We really don't have a single approach to anything. Different schools and majors probably approach the topic differently. My college had a linear algebra course specifically crafted for CS majors and engineers. I took that and it did focus on matrices. It was also the only math class that required programming. I believe math majors had their own linear algebra course.
> My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.
Different strokes for different folks. If it worked out for you that's all that matters.