(The first sentence is the most important in my opinion)
“A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied (“scaled”) by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field*. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms [(listed in § Definition)]. For specifying that the scalars are real or complex numbers, the terms real vector space and complex vector space are often used.”
“A vector space over a field* F is a set V together with two [certain] operations…
- “The first operation, called vector addition or simply addition… takes any two vectors v and w [in V]and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. (The resultant vector is also an element of the set V.)
- “The second operation, called scalar multiplication… takes any scalar a [in F] and any vector v [in V] and gives another vector av. (Similarly, the vector av is an element of the set V. Scalar multiplication is not to be confused with the scalar product, also called inner product or dot product, which is an additional structure present on some specific, but not all vector spaces. Scalar multiplication is a multiplication of a vector by a scalar; the other is a multiplication of two vectors producing a scalar.)”
Quotes taken from “Vector space,” Wikipedia, retrieved 6/10/2020
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*What is the Gist of a “Field” (in mathematics)?
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Disclaimer:
I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).
