Vector space
Definition 1.1.1. Let 𝕂 be a field, whose elements are referred to as scalars. A vector space over 𝕂 is a nonempty set V, whose elements are referred to as vectors, together with two operations. The first operation, called addition and denoted by +, assignsan element u + v ∈ V to each pair of elements u, v ∈ V. The second operation, called scalar multiplication is a function that assigns an element λv ∈ V to each λ ∈ 𝕂 and each v ∈ V. Furthermore, the following properties must be satisfied:
Commutativity of addition: u + v = v + u
Associativity of addition: (u + v) + w = u + (v + w).
Existence of a zero: There is a zero vector 0 in V such that 0 + u = u + 0 = u.
Existence of additive inverses: For each u ∈ V , there is a vector u in V such that u + (−u) + u = 0.
Closure under addition: The scalar multiple of u by λ, denoted by λu, is in V .
λ(u + v) = λu + λ v.
(λ1 + λ2) u = λ1 u + λ2u.
λ1(λ2u) = (λ1λ2) u.
1 u = u
Note that the first four properties in the definition of vector space can be summarized by saying that V is an abelian group under addition.
A vector space over a field 𝕂 is sometimes called an 𝕂-space. A vector space over the real field is called a real vector space and a vector space over the complex field is called a complex vector space.
We see now some examples of vector spaces.
Definition 1.1.2 Suppose n is a positive integer. A list of length n (or n-tuple) is an ordered collection of n elements (which might be numbers, other lists, or more abstract entities) separated by commas and surrounded by parentheses. A list of length n looks like this:
(x1, ..., xn)
Two lists are equal if and only if they have the same length and the same elements in the same order.
Remark. By definition each list has a finite length that is a positive integer. Thus an object that looks like (x1, x2, ...), which might be said to have infinite length, is not a list. A list of length 0 looks like this: ().
The set 𝕂n of all ordered n-tuples whose components lie in a field 𝕂, is a vector space over 𝕂, with addition and scalar multiplication defined componentwise.
(x1, ..., xn) + (y1, ..., yn) = (x1y1, ..., xn + yn)
and
λ(x1, ..., xn) = (λx1, ..., λxn)
Examples 1.1.3. Sono spazi vettoriali:
Let 𝕂 be a field. The set 𝕂𝕂 of all functions from 𝕂 to 𝕂 is a vector space over 𝕂, under the operations of ordinary addition and scalar multiplication of functions:
(f + g)(x) = f(x) + g(x)
and
(af)(x) = a(f(x))
The set Mn,m(𝕂) of all n x m matrixes with entries in a field 𝕂 is a vector space over 𝕂, under the operations of matrix addition and scalt multiplication.