I'm just wondering what are the differences between vector spaces and fields. From what I understand by the definitions, both of these are collections of objects where additions and scalar multiplications can be performed. I can't seem to see the difference between vector spaces and fields.
You can multiply the elements of a field together. Generally you cannot do this with a vector space, but you can multiply elements of a vector space by elements from the underlying field. In this way, every field is a vector space over itself. In fact, every field is a vector space over any of its subfields.
This is incorrect. "scalar multiplication" is not defined in a field, multiplication of two members of the field is defined. A vector space has to be defined "over" a specific field, in which case members of the field are the 'scalars'. You cannot, in a general vector space, multiply two vectors nor can you divide by a (non-zero) vector. You can multiply any two members of a field and divide by a non-zero vector.
an example of a vector space is the set of arrows beginning at the origin in a plane with chosen origin. you add them by the parallelogram law, and multiply them by rational numbers by adding them to them selves and subdividing them into equal pieces. but you are not given any way to multiply two arrows together.
I had the same the question originally posed in this thread. I now understand why vector spaces and fields in general are distinct algebraic structures, however, I am still curious about what vector spaces are used for exclusively that other algebraic structures aren't?
Essentially, vector spaces, because they have "scalar multiplication" but not multiplication of two vectors, model "linearity" where other algebraic structures don't. Any situation, such as systems of linear equations, or linear differential equations, in which "linearity" is important can be reduced to vector spaces.
That makes a lot of sense, thanks! So, am I now correct in thinking that vector spaces model linearity so well, because things that are being scalar multiplied depend on elements from another set?
A field is a vector space over itself. So multiplication of two field elements is also scalar multiplication.
I see now that I may have misunderstood the question. I was reading it as "What is the difference between a 'vector space' and 'field'" but I think now it was "What is the difference between a 'vector space' and 'vector field'. Of course, 'field' and 'vector field' are quite different things. A "vector field" is a geometric situation in which we have a geometric object such as a manifold and, at each point of that object, defines a vector (we can have a vector space at each point on the object) .
* A field (in the abstract algebra sense, not the same thing as a vector field, for which see below) consists of two commutative groups with the same underlying set--except that the identity element with respect to the group operation of the 1st group (called addition) has no inverse with respect to the group operation of the 2nd group (called multiplication)--and multiplication distributes over addition thus: a(b + c) = ab + ac = (b + c)a. * A vector space consists of a commutative group, elements of whose underlying set, V, are called vectors, a field, as defined above, elements of whose underlying set, F, are called scalars, and a function, s, called scalar multiplication, s: F x V --> V, such that, if we denote it by juxtaposition of a scalar and a vector, and use bold letters for vectors and ordinary Roman letters for scalars: (1) a (b u) = (ab) u; (2) a(u + v) = au + av; (3) (a + b)u = au + bu; (4) 1u = u. The notation ua is taken to be equivalent to au. A vectors space is said to be "over" its field, e.g. a vector space over the reals, i.e. over the field of real numbers. A particular vector space may have further structure defined on it, such as an inner product, in which case it can be called an inner product space. But whatever extra structure it has, it will still be a vector space if it meets the above requirements. A familiar example is the vector space over the reals whose vectors are elements of R^{n}, the set of sequences of real n numbers, called n-tuples, with addition defined componentwise: (a_{1},a_{2},...,a_{n}) + (b_{1},b_{2},...,b_{n}) = (a_{1} + b_{1}, a_{2} + b_{2}, ..., a_{n} + b_{n}). * A vector field (no relation to algebraic field, the sameness of the names is an unfortunate coincidence) is a function F: U --> V, where U is some subset of the underlying set of a manifold, and V the underlying set of a vector space, which associates each point in U with a vector from V. If you haven't met manifolds yet, their definition takes a bit of work, but don't worry about that for now: a simple example is Euclidean space of some dimension. A vector field on that Euclidean space (usually assumed to be on the whole of Euclidean space unless otherwise stated) is a function that associates a particular vector with each point of Euclidean space. We think of there being a separate vector space at each point, called the tangent space at that point. A vector field selects one vector from each tangent space. Vector fields are often depicted in diagrams showing arrows are extending from a bunch of representative points. A well known example is the kind of weather map that shows a vector field of wind velocity vectors as a scatter of little arrows pointing in the direction of the wind and representing its strength by their length.