TITLE: Are groups and rings more difficult algebraic structures to understand than vector spaces?
QUESTION [7 upvotes]: I have read multiple posts on here and in other places where most people seem to recommend to learning linear algebra before abstract algebra. Is that because vector spaces are simpler to understand than groups and rings? I am having some challenges with understanding how certain aspects of vector spaces work, I was wondering if learning about rings and/or groups can help me better understand how vector spaces work?

REPLY [4 votes]: Linear algebra can be viewed with some advanced algebraic tools; for example a vector space can be thought of as an abelian group paired with an additional operation of scalar multiplication. We can consider vector spaces over finite fields. Matrices can be viewed as providing homomorphisms between vector spaces. Furthermore, groups of invertible matrices provide important examples of nonabelian groups.
On the other hand, a first introduction to linear algebra can be given without looking too deeply at the algebraic machinery. Students don't need to learn really any group theory or field theory to consider vector addition over the real or complex numbers. We can teach students about linear maps between vector spaces without going too deeply into the concept of homomorphisms between algebraic structures.
In short, linear algebra before "abstract algebra" is often recommended because an introductory look at linear algebra provides a good first look at concepts that will come up in a more advanced algebra course. It is particularly nice to teach students about nonabelian groups when they are already familiar with matrix multiplication, as this allows lots of interesting examples to be presented early on in the course.
BUT: Linear algebra is a really beautiful and deep subject. There are lots of cool advanced topics that can be covered once you are familiar with some advanced topics in algebra. So it is definitely worth coming back and taking a second course in linear algebra once you do understand groups, rings, fields, modules, homomorphisms, etc.