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A Vector space is a set of Vectors, V, a set of scalars (known as a Field) as well as the rules for Vector addition, and the multiplication of a Vector by a scalar.

(V,F,+,*)

Examples of sets of vectors:

set of Polynomials 

 ℝn (n euclidean  space) the plane is ℝ2

the set of all nxm, Matrices (where n and m are constant).

Examples of sets of Fields:

ℝ (the set of real numbers)

ℂ (the set of complex numbers)

Example of the rule for vector addition in ℝ2:

v, w ∈ℝ2 ie

v = [x1, x2], w = [y1,y2]

v+w = [x1 + y1, x2 + y2]

Example of the rule for vector multiplication in ℝ2:

v∈ ℝ2 , z ∈ ℝ

v = [x1,x2]

A vector space consists of a field (a set with well-defined addition, subtraction, multiplication, and division, like the rational, real, or complex numbers) and a set of vectors (if you think of them as vaguely undefined elements that can get scaled, you are golden. If you think of them as an ordered set of numbers, know that you are only thinking of a small but very important subset of vectors).

There are a few rules. First, if u and v are vectors, you can add them. Second, if a is a “number” in your field (we call these scalars), then av is also in your vector space. We do not need to define a multiplication of vectors (in some special cases we can).

Using ai from our field and vi which are vectors, we can make the linear combination a1v1+a2v2+…+anvn. Because of our rules, this in in our vector space.