/sci/ - Science & Math

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>>9542728
>>9542730
>A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below.

>The first operation, called vector addition or simply addition + : V × V → V, takes any two vectors v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. (Note that the resultant vector is also an element of the set V ).

>The second operation, called scalar multiplication · : F × V → V, takes any scalar a and any vector v and gives another vector av. (Similarly, the vector av is an element of the set V ).

>Elements of V are commonly called vectors. >Elements of F are commonly called scalars.

>In the two examples above, the field is the field of the real numbers and the set of the vectors consists of the planar arrows with fixed starting point and of pairs of real numbers, respectively.

>To qualify as a vector space, the set V and the operations of addition and multiplication must adhere to a number of requirements called axioms.[1] In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F.