Vector space, linear combination and vector field I just learnt about vector space and I want to make sure I get the concept right.


*

*Vector space/linear space is a type of space in which there are vectors, made up or basis vectors that match the coordinates of the vector space. 


2.'Vector space over a field' means vectors(made up of basis vector which matches the coordinates of the vector space) are scaled by a scalar in a field(a set of elements with algebraic operations). The value/expression of the linear combination of all vectors thus describes the characteristics of the vector space? 


*Is this linear combination description of a vector space the vector field, the values of which are defined by the scalar field? 


I am pretty new to maths so just want to make sure I get the basics. 
 A: 1.
It seems like you may be trying to understand vector spaces for the first time from multiple sources at once, which I would not recommend. Depending on the book/introduction:

*

*The spaces could be called "vector space" or "linear space"

*Vector spaces could always have a collection of "basis vectors" or not necessarily.

*Vectors in a vector space could always be written with coordinates, sometimes be written with coordinates, or not always be possible to be written with coordinates.

As DonAntonio alluded to in a comment, I would recommend you get one source text (whether it's online or not) and work through the beginning of it. Then later you can learn what is similar/different about different treatments.
2.
I'm not sure exactly what you're claiming here, so I don't know how to answer, but "linear combination of all vectors" is not something that usually arises.
Briefly, a field is a bunch of numbers, like the real numbers $\mathbb R$. A "vector space over a field" means you can scale vectors by numbers (called "scalars") in that field.
A "linear combination" just means a sum of scaled vectors. Like if $\mathbf x$ and $\mathbf y$ and $\mathbf z$ are vectors, then $\mathbf x+2*\mathbf y-3*\mathbf z$ is a linear combination of those three vectors.
The characteristics of a vector space is not formally defined, but I would say depends at least in part on which vectors are equal to linear combinations of which other vectors.
3.
The use of "field" in "vector field" and "scalar field" is unfortunately completely different from the use of "field" in "vector space over a field".
