Are groups and rings more difficult algebraic structures to understand than vector spaces? 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?
 A: Yes, generally speaking vector spaces are particulary simple algebraic structures. Therefore, they make a fine introduction to more advanced aspects of Abstract Algebra.
For instance, if you are working with vector spaces over, say, the real numbers, and if $v$ is a vector, then you never have$$\overbrace{v+v+\cdots+v}^{n\text{ times}}=0\tag1$$(unless $v=0$). Over some other fields (say, finite fields), there is a natural number $n$ such that you always have $(1)$. In a group (or a ring), you can have $(1)$ for certain elements and certain $n$'s, whereas for other elements you don't have $(1)$, no matter which $n$ you choose.
And every vector space has a basis. The natural generlization of vector spaces over a fields are modules over a ring. And these seldom have a basis.
Furthermore, there is a very simple classification of all vector spaces over a filed: up to isomorphism, for each cardinal there is one and only one vector space whose dimension is that cardinal. There's nothing similar for grous or rings.
A: 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.
A: Yes. Your understanding is correct. There is a theorem that any two finite dimensional $k$-vector spaces (vector spaces defined over the field $k$) of the same dimension are isomorphic:
$$ \dim(V_1) = \dim(V_2) = n < \infty \implies V_1 \simeq V_2 \simeq k^n.$$
However, this is not the case for groups and rings. In fact, two finitely generated abelian groups may have the same rank but not be isomorphic: the obvious example is in rank 0, considering the Klein four group
$$
\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \not\simeq \mathbb{Z}/4\mathbb{Z}.
$$
(Recall that finitely generated abelian groups may be expressed in the form
$$
 G = \mathbb{Z}^r \oplus G_{\text{tors}}
$$
where $r$ is the rank and the torsion part $G_{\text{tors}}$ is finite.)
And that is just in the finitely generated abelian case! Loosening these assumptions, there are far more possibilities to contend with. So it makes sense to start by studying the simpler case of linear algebra.
A: I learned group theory and ring theory before learn about vector spaces and linear algebra in general. As I see these three subjects built for three different purposes, at least at the beginning. Group theory to study general solutions of algebraic equations (basically Galois theory), ring theory for solutions of system of polynomial equations (algebraic geometry), and linear algebra for solutions of system of linear equations. So, while being algebraic theories, they have different structures and different flavors. Of course learning any of theses first would be beneficial to understand the other, but I won't say one is particularly simple or interesting than others.
A: I don't think so.  There is a certain amount of overlap, and both subjects range from difficult and complicated to simple and trivial.
For one or two examples, a vector space can be viewed as an abelian group with some additional structure. Furthermore, any abelian group is a $\Bbb Z$-module.   And, any field is a vector space over its prime subfield.
I noticed that there is at least one book that treats both subjects simultaneously.  Linear Algebra and Group Theory, by V.I. Smirnov.
