Abstract Algebra can be considered as component of linear algebra in the sense that the
sets that linear algebra are built around (Abelian groups) are a mainstay of Abstract Algebra. In
some textbooks, it seems that Abstract Algebra begins with Groups and continues through Rings
and Fields. In other earlier books (Modern Algebra by Birkhoff and MacLane and Topics in
Algebra by Herstein), it seems that the idea is that the abstraction of “Algebra” embraces
everything through Linear Algebra. I have even read that Abstract Algebra can “subsume” linear
algebra.
Whichever is the case, there is no doubt that Vector Spaces, a main component of Linear
Algebra, uses an Abelian Group for its vectors. Nor is there any doubt that Matrices, another
topic of Linear Algebra can be viewed as an Abstract Algebra topic in large part, if one restricts
one’s study to the addition and subtraction of matrices.
The question of “how to shorten the list of properties that define a vector space by using
definitions from abstract algebra” is ambiguous. My understanding of vector spaces is that there
are eight definining conditions on the pairing of two sets, a vector space V and a field F, with
two operations: + and ·.
The first four conditions listed in most texts relate to the operation + over the elements of
V (“addition” is the word traditionally used for this operation +):
1. That + is associative over V
2. That + is commutative over V
3. That V contains a unique additive identity, traditionally called “0”.
4. That each element of V has a unique additive inverse. (for v in V, v is the designation of its
inverse).
Additionally, there are four more rules that describe the interactive behavior of + and · .
If we use Greek letters for the members of F and roman letters for the members of V,
contiguousness to represent “multiplication” (· ), and 1 to represent the multiplicative identity,
then these rules are (for all  ,   F and a, b V ):
5.  (v  w)   v   w
6. (   )v   v   w
7.  (vw)  ( v) w
8. 1v  v
The first 4 components of the definition of a vector space in older Abstract Algebra texts
are not itemized as four rules. In both the Herstein and Berkhoff textbooks, there is simply the
statement that V is an Abelian Group under +. It is this that I see as a “shortening of the list of
properties for a vector space.” It isn’t really, because one way or another, those four properties
must exist. Calling those properties an Abelian Group simply hides those four properties as a
single description. But Abelian Group has to be defined somewhere, and wherever that
definition exists, those four properties will be present.
What I get a sense of if that more than not, Linear Algebra and Abstract Algebra are more
neighbors, in a sense, than one the subordinate of the other. Both seem to be part of a large
thinking structure called simply Algebra or, in the case of one book, “Modern Algebra”. I’ve
read that as one develops the abstractions arising from Linear Algebra that there comes a point
where these components give “a complete description and construction of all finite Abelian
Groups.” (p. 130, Topics in Alebra, I. N. Herstein, Blaisdell Publishing Company, 1964).