let's talk about what a field is first. i'll try not to be too technical, but understand that being technical is also being more precise, so some of my definitions will be a bit vague.
to explain what a field is, we need a more basic idea, that of a binary operation on a set.
a binary operation on a set S, is a function from pairs in S to S. like this:
f(a,b) is in S. so you start with two things in S, a and b, and you combine them to get some third element of S.
a field has two of these operations, + and * (addition and multiplication). these have to obey the following rules:
A1) (a+b)+c = a+(b+c) (addition is associative)
A2) a+b = b+a (addition is commutative)
A3) there is an element, usually written 0, in F with
a+0 = 0+a = a (existence of an additive identity)
A4) for every element a, there is another element
usually written -a, with a+(-a) = (-a)+a = 0 (existence of inverses)
M1) (a*b)*c = a*(b*c) (multiplication is associative)
M2) a*b = b*a (multiplication is commutative)
M3) there is an element, usually written 1 ≠ 0, with
a*1 = 1*a = a (existence of multiplicative identity)
M4) for all a ≠ 0, there is another element, usually written 1/a,
with a*(1/a) = (1/a)*a = 1
as you can see we have two kinds of stuff we can do in F, add and multiply. but for F to be a field, we need more, we need a rule that relates how addition and multiplication interact. this is:
D1) a*(b+c) = a*b + a*c (distributive law)
in simple terms, fields are things in which all the usual laws of arithmetic are true. for example, the set of all rational numbers (fractions), and the set of all real numbers are both fields. the set of all integers is NOT a field, because (M4) is not true (1/4 is not an integer).
now, vector spaces are usually "bigger" than fields. and they have their own set of rules.
vectors have their own kind of addition, called "vector addition", which has to satisfy rules A1 through A4.
where vector spaces get sorta weird, is that there is a special way to multiply vectors by a field element (or a scalar, called that because they "scale" things).
so if a is in our field, and v is a vector, we can form a●v, as another vector.
one vector space that is familiar to many people is the following:
the plane, consisting of points (x,y) where x and y are real numbers.
for two vectors, (x1,y1) and (x2,y2) we define the "vector sum" to be
(x1,y1) ┼ (x2,y2) = (x1+x2, y1+y2)
(so (3,4) ┼ (-2,7) = (1,11), for example).
(i am using ┼ here to emphasize that although ┼ obeys the same rules as +,
it is not the same thing).
and the "scalar product" for a real number a, and a vector (x,y) to be:
a●(x,y) = (ax,ay) (so 2●(5,-4) = (10,-8), for example).
vector spaces have their own rules besides just A1-A4 for ┼.
V5) a●(u┼v) = a●u ┼ a●v
V6) (a+b)●u = a●u ┼ b●u
V7) a●(b●u) = (ab)●u
V8) 1●u = u
these are all there to make sure that vector addition and scalar multiplication are "compatible".
an inner product space, is a special kind of vector space, with an inner product. as you may have guessed, inner products have even more rules (which i won't go into here). what i will say, is that inner products take two vectors, and combine them to get a scalar (field element).
for example, in the plane, you can define an inner product of (x1,y1) and (x2,y2) as
(x1,y1)·(x2,y2) = (x1)(x2) + (y1)(y2).
so the inner product of (3,2) and (-2,4) would be:
(3)(-2) + (2)(4) = -6 + 8 = -2.
inner products can be used to define the length of a vector, and the angle between two vectors, which gives a link between geometry and linear algebra (the study of vector spaces).