# Difference between a vector space and a field?

by wk1989
Tags: difference, field, space, vector
 P: 32 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.
 Sci Advisor HW Helper P: 2,020 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.
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PF Gold
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 Quote by wk1989 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.
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.

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## Difference between a vector space and a field?

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.
 P: 2 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?
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,705 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.
 P: 2 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?
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 Quote by HallsofIvy This is incorrect. "scalar multiplication" is not defined in a field, x
A field is a vector space over itself. So multiplication of two field elements is also scalar multiplication.
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