Appendix A Vectors and Vector Spaces A vector can be defined in - PDF document

Appendix A Vectors and Vector Spaces A vector can be defined in many ways: it can be interpreted a collection of real or complex numbers arranged in a row or in a column, it can represent directed, i.e., geometrical quantities, as forces, moments, velocities, etc., it is an element of a linear vector space, with many algebraic properties that can be put into correspondence with geometrical properties. In the present notes we will introduce some of the above definitions, without losing our aim, that is to empower the students to use vectors as a basic tool for dynamical systems modelling; so while we will avoid any superfluous information, we will try to be formally correct, but omitting any theorem or proof, apart from those that give an insight to the underlying physical interpretation. Let us start with the most common definition: a vector is a collection of n real or complex quantities arranged in a single column   v 1 v 2     v = .  .  .   v n So a vector can also be seen as a particular matrix, with dimension n × 1. As anticipate in Section 1.3, if we need to reduce the printed space, we can write the vector as a transpose column, i.e., a row [ ] T v = v 1 v 2 · · · v n In broader terms, a vector is an element of a vector space; the following Section will review some concepts and definitions on Vector Spaces. 163

164 Basilio Bona - Dynamic Modelling A.1 Vector spaces A vector space is a mathematical structure whose elements, denoted by v , obeys to a number of rules that qualify the properties of vector spaces. These properties are well suited to describe a large number of phenomena characterizing many sectors of physics and engineering; for further details see [8]. Definition Given a generic field 1 F , a vector space is a set of elements, called vectors , that satisfy the following axiomatic properties 1. A vector sum operation, denoted by +, exists, such that { V ( F ); + } is a commutative (or abelian) group; the identity element is denoted by 0 ; 2. For any α ∈ F and any v ∈ V ( F ), a vector α v ∈ V ( F ) exists; moreover for any α, β ∈ F and any v , w ∈ V ( F ) the following are true: (a) associative property with respect to the product by a scalar: α ( β v ) = ( αβ ) v (b) existence of the identity with respect to the product by a scalar: 1( v ) = v ; ∀ v (c) distributive property with respect to the vector sum: α ( v + w ) = α v + α w (d) distributive property with respect to the product by a scalar: ( α + β ) v = α v + β v A vector space is said to be real if F = R , while is said to be complex if F = C . A classical example of a real vector space is that whose elements are collections of n real numbers V n ( R ) = R n ; in this case an element, i.e., a vector, is represented by components   v 1 v 2      , v ∈ R n , v i ∈ R v = .  .  .  v n 1 See Appendix C for this and other abstract algebra structures.

165 Basilio Bona - Dynamic Modelling Since the properties (a)–(d) induce a linear structure on the space V , this is also termed as linear vector space or more simply linear space . The term “vector” therefore takes a meaning that is more general and can define ab- stract mathematical entities more generals that the simple collection of n numbers. For instance, as shown in [30], infinite sequences of real or complex numbers, con- tinuous functions taking their values in the interval [ a, b ], polynomials with complex coefficients defined in [ a, b ], oriented segments ⃗ v , and many more. As one can notice, among the vector space axioms, no product operator is defined. For this reason the vector space structure , i.e., the set of properties deriving from the above axioms, does not allow to define such geometrical concepts as angle or mag- nitude . that are implicit in a purely geometrical definition of the directed quantity ⃗ v . To allow a definition of these geometrical concepts it is necessary to endow the vector space with a quadratic structure also know as metric structure or simply metric . The introduction of a metric structure on a vector space generates an algebra that makes possible the performance of a number of geometric computations involving the vectors. The most common metric is that induced by the scalar product definition. Before defining this product, it is convenient to review some properties of the linear functions. A.2 Linear functions and operators Given two vector spaces U ( F ) and V ( F ), that for simplicity we assume defined on the same field F , a function L : U → V that transforms the elements in U in elements of V , is said to be linear if for couple of vectors a , b ∈ U and any scalar λ ∈ F , the following axioms hold true L ( a + b ) = L ( a ) + L ( b ) = L a + L b (A.1) L ( λ a ) = λ L ( a ) = λ L a The function L : U → U is also called linear operator , linear transformation , linear application or endomorphism . The set of all linear functions L : U → V is a linear vector space L ( F ). Moreover the set of all linear functions L : U → U is a ring 2 denoted by the symbol End( U ). We also point out that any linear transformation from U to V can be represented by a matrix A ∈ R m × n , and vice versa, where m and n are the dimensions of V and U . 2 See Appendix ?? . completare

166 Basilio Bona - Dynamic Modelling Linear independence – Base – Dimension Given n vectors a i ∈ V ( F ), a vector v ∈ V ( F ) is said to be a linear combination of { a 1 , a 2 , . . . , a n } if it can be written as v = v 1 a 1 + v 2 a 2 + · · · v n a n (A.2) with v i ∈ F . The vector set { a 1 , a 2 , . . . , a n } is said to be linearly independent if no a i can be written as a linear combination of the remaining ones a j , j ̸ = i . In other words, the only solution to the equation   0 . def   . v 1 a 1 + v 2 a 2 + · · · + v n a n = 0 = .   0 is that with all v 1 = v 2 = · · · = v n = 0. If this is not the case, we say that a i is linearly dependent from the other vectors. In the linear combination v = v 1 a 1 + v 2 a 2 + · · · v n a n if all vectors a i are linearly independent, then the scalars v i are unique and take the name of coordinates or components of v . The linear combinations of k ≤ n linearly independent vectors { a 1 , a 2 , . . . , a k } form a subspace S ( F ) ⊆ V ( F ). It is said that this subspace is spanned by { a 1 , a 2 , . . . , a k } . Any set of linearly independent vectors { a 1 , a 2 , . . . , a n } forma basis in V . All bases in V have the same number of elements (in our case n ); this number is the dimension of the space and is denoted as dim( V ). Often, in modern physics textbooks, such as those on A note on notations relativity, the relation in (A.2) is written as v = v 1 a 1 + v 2 a 2 + · · · v n a n (A.3) simply written as v = v i a i or v = v i a i that makes use of the Einstein conventions for the sum n ∑ def v i a i v i a i = i =1 Indeed in the tensorial representation a vector a can be expressed in two different ways, adopting the Einstein convention: a = a i e i a = a i e i oppure

167 Basilio Bona - Dynamic Modelling where a i are the so-called contravariant components of a , while a i are the so-called covariant components of a , e i are the contravariant basis vectors and e i are the ( covariant basis vectors ). In these notes we will adopt the column vector notation introduced previously, but we suggest the reader to familiarize with different notations adopted in different contexts. A.3 Vectors and their interpretation Once the concept of vector space has been defined, it is important to briefly discuss the interpretation associated to vectors, since this can be useful to represent both geometrical and physical entities for modelling purposes. Geometrical vectors A geometrical vector p represents a point P in space. The point P is an abstrac- tion that often, but not always, requires a representation. Vector representations are given with respect to a reference frame. If P ∈ R 3 then its representation is a vector     p x p 1 p a ∈ R 3 = p y ≡ p 2     p z p 3 a a and p i is the vector i -th coordinate with respect to a chosen reference frame R a . Affine geometry To treat points as vectors in linear vector spaces implies the definition of a zero point (the origin), i.e., a point with particular privileged characteristics. Since, in many applications, this is not required, a particular geometry that is “origin-free” has been set up. It is called affine geometry and is defined on affine spaces . Affine geometry is at the base of projective geometry and perspective transforms, as well as homogeneous vectors and homogeneous transformations. It will not be considered in the present context; the interested reader can find additional details in [14]. Approccio geometrico TO BE COMPLETED TRADURRE