Differences Between Vector Spaces and Fields Explained

[UrbanXrisis]

What is the different between a vector space and a field? Seems to me that they both are the same thing

 

[cristo]

What is the different between a vector space and a field? Seems to me that they both are the same thing

A field is a set with addition and multiplication defined between elements of the set (along with other axioms). A vector space (over a field) is a set with addition defined between elements of the set, and multiplication defined between an element of the field and an element of the set.

That is, in a vector space, there is no notion of multiplication between vectors.

 

[HallsofIvy]

What is the different between a vector space and a field? Seems to me that they both are the same thing

Then I suggest that you look at the definitions again! A field has two operations defined- multiplication and addition. Each combines two members of the field and gives a member of the field as a result. In particular, there exists a multiplicative identity and every member of the field except 0 must have a multiplicative inverse.

In a vector space, we have addition defined as combining two vectors and resulting in a vector but the "multiplication" (scalar product) combines a vector and a member of the underlying field (every vector space must be defined "over a field") and results in a vector. Because we do NOT multiply two vectors, It doesn't even make sense to talk about a "multiplicative inverse" for a vector space.

For example, the set of all polynomials of degree two or less is a vector space over the real numbers but is not a field.

Given a field, it is possible to use its addition and multiplication to think of it as a (one dimensional) vector space over itself but there always exists many other vector spaces over the same field.

 

[pkokeny]

I believe the term "vector field" can be misleading. It refers to a vector valued function that outputs an n-dimensional vector to every point in some n-dimensional space. So when the word "field" is used in that context, it better conceptualized as a force field (like gravity in 3 dimensions), not a field of scalars, as described in the current replies.

With that being said, no, a field and a vector space are not the same thing. A vector space is a "space" that includes all possible vectors, in n-dimensions, over some field of scalars. These vectors can be added and subtracted, but not multiplied because they are not numbers! They use numbers (real numbers are an example of a field) to give them some magnitude and direction in n-dimensional space... But a vector field, as described above, actually "reveals" (for lack of a better term) whatever vectors in the vector space that are outputted by the vector valued function that is that vector field.