Let S be a non-empty set. A set is just a collection of elements with no structure. You can't do anything interesting with them. It isn't until we equip S with a binary operation (addition, multiplication, composition, convolution etc.) defined by *:SxS -> S such that (a, b) maps to a*b under the operation that we can start to say anything meaningful about S. When we have such a rule, we look at the way * operates on the elements of the set. We can ask a few interesting questions:
Is a*b in S for every combination of a and b in S?
Is there some special element (say, e) so that a*e = a = e*a for every element a in S?
Is there a special element a' in S so that a*a' = a'*a = e in S?
What about equipping S with more than one binary operation?
A vector space V is such a structure. A vector space is a set V equipped with multiplication (*) and addition (+) so that we have what amounts to two separate structures on V with respect to each binary operation. The full tuple that defines a vector space V would be (V, +, *, 0, 1, k) where V satisfies:
For (V, +, 0):
1) u + v is in V for every combination of u and v in V.
2) u + v = v + u (which isn't always true for every binary operation. Think matrix multiplication under (*))
3) There exists a u' in V called the "inverse" of u so that u + u' = u' + u = 0.
4) Since 0 is in V, u + 0 = 0 + u = u for every u in V.
And for the structure with respect to (*), we have for (V, *, 1, K) where K is the field over which the vector draws its scalars... (think R. It will almost always be R for your level), we have:
5) u*v is in V for every combination of u, v in V
6) u*(m + n) = u*m + u*n for every u in V, n,m in K. This is just your normal distributivity wr.t. vectors in V. (likewise, (m + n)*u = m*u + n*u. Take care to keep your vectors on the side with which you're multiplying)
7) (u + v)*m = m*(u + v) = m*u + m*v for every u, v in V, m in K.
8. (m*n)*u = m*(n*u) for every u in V, m,n in K.
9. 1*u = u*1 = u for every u in V.
10. 0*u = u*0 = 0 for every u in V.
Now, this may seem confusing, and it certainly is if this is your first time seeing axioms that define a structure, but it makes tons more sense if you suss it out thusly: "When i have a set and an operation on that set, what meaningful things can i do with that binary operation that preserve the structure of that set with respect to that operation?" This is the point of Algebra in general - specifically what you begin to touch upon in the early years of linear algebra. We never stray outside of our set. We always maintain the structure of the set with respect to the binary operations we're given.
Vectors in and of themselves do not exist outside of a vector space, and they're important because it allows us to orient ourselves arbitrarily in space on given basis (you'll learn this eventually). Right now, memorize.
