Overview

• Motivation: systems described by multiple state parameters

• Basic concepts: scalars, vectors, matrices

• Sets: finite, countable, countably infinite, uncountably infinite

• Algebraic structures: groups, fields, field of real numbers

• Functions: general definition, vectors as functions defined on a finite set

• Up to now, systems described by $y:ℝ\to ℝ$, $y^{\text{'}}=f(x,y)$ have been considered

• Most systems require more than one parameter to establish their state, e.g.,

  • Position in 3D-space: $r\left(t\right)=\left(\begin{array}{lll}x\left(t\right)& y\left(t\right)& z\left(t\right)\end{array}\right)\in {ℝ}^3$, $r:ℝ\to {ℝ}^3$;

  • Stock market prices: $p:ℝ\to {ℝ}^n$, $n$ the number of listed stocks,

  • Patient health monitoring: $y=\left(\begin{array}{lll}\mathrm{temperature}& \mathrm{heart}\mathrm{rate}& \dots \end{array}\right)$

• Change in system whose state is given by $n$ parameters that each depend on a single independent variable $x$

  ${\displaystyle y^{\text{'}}=f(x,y),y:ℝ\to {ℝ}^n,f:ℝ\times {ℝ}^n\to {ℝ}^n}$

  (1)

• The mathematical issue that arises is to establish a framework to work with multi-dimensional quantitites, such as $y\left(x\right)$ from (1)

• A scalar is a quantity described completely by its magnitude. Such quantities are described mathematically by a single number such as $n\in \mathbb{N}$ or $\alpha \in ℝ$

• Informally a vector is a quantity that besides magnitude also requires the specification of a direction. This usage arises from common examples in physics, e.g., the position vector of a point $\stackrel{\rightharpoonup }{r}=x\stackrel{\rightharpoonup }{i}+y\stackrel{\rightharpoonup }{i}+z\stackrel{\rightharpoonup }{k}$. Consider for now a vector to be a grouping of multiple scalars, e.g.,

  ${\displaystyle 𝒙=\left(\begin{array}{l}{x}_1\\ ⋮\\ x_m\end{array}\right)\in {ℝ}^m.}$

  A mathematically precise definition is given later.

• Just as scalars are grouped into vectors, $n$ vectors are grouped into a matrix

  ${\displaystyle 𝑨=\left(\begin{array}{lll}{𝒂}_1& \dots & {𝒂}_n\end{array}\right)\in {ℝ}^{m\times n},{𝒂}_1,\dots ,{𝒂}_n\in {ℝ}^m.}$

• The informal introduction of vectors, matrices can lead to contradictions. A consistent mathematical framework is required. First step: naïve set theory.

• In naïve set theory

  1. The resulting set theory is called “naïve” because it can lead to paradoxes, e.g., the Russell paradox: “What is the set of all sets that are not members of themselves”. Mathematically: define $R=\left\{x|x\notin x.\right\}$. Then the paradox is that $R\in R\iff R\notin R$, i.e. if $R$ is a member of $R$ then it is not a member and if $R$ is not a member of $R$ then it is a member. Such paradoxes led to the formulation of axiomatic set theory.

  ¹ , a set is a collection of objects , e.g., $A=\{a,2,Z\}$ , and a fundamental binary relation between objects and sets is introduced, denoted by $\in$ , the member of relation , such as $2\in A$ . The negation of this relation is $\notin$ , the not member of relation , e.g., $3\notin A$ .

• Some sets are finite, e.g., $A=\{a,2,Z\},B=\{1,2,\dots ,n\}$

• Other sets are infinite, but countable: $\mathbb{N}=\{0,1,2,\dots \}$,${\mathbb{N}}^{\ast }=\{1,2,3,\dots \}$, $\mathbb{Z}=\{0,\pm 1,\pm 2,\dots \}$, $\mathbb{Q}=\{a/b|a\in \mathbb{Z},b\in {\mathbb{N}}^{\ast }.\}$

• Yet other sets are infinite and not-countable: $ℝ$, $ℂ$

• After introducing objects and collections of objects in set theory, the next step is to formally define operations with objects. Algebra defines many such formal structures. Of particular interest here are groups and fields.

• A group is the algebraic structure $(𝒮,\oplus )$ with $𝒮$ a set, $\oplus$ a binary operation with properties for $\forall a,b,c\in 𝒮$:

  Closure. $a\oplus b\in 𝒮$.

  Associativity. $a\oplus (b\oplus c)=(a\oplus b)\oplus c$.

  Identity element. $\exists e\in 𝒮$, such that $a\oplus e=e\oplus a=a$

  Inverse element. $\exists (-a)$ such that $a\oplus (-a)=e$

• In addition to the above properties, a commutative group also satisfies:

  Commutativity. $a+b=b+a$.

• A field is an algebraic structure $(𝒮,\oplus ,\otimes )$ with commutative group properties for $\oplus$ and $\otimes$ (with an exception) and distributivity of $\otimes$ over $\oplus$, $\forall a,b,c\in 𝒮$

  $a\oplus b\in 𝒮$	$a\otimes b\in 𝒮$
  $a\oplus (b\oplus c)=(a\oplus b)\oplus c$	$a\otimes (b\otimes c)=(a\otimes b)\otimes c$
  $\exists 0\in 𝒮$, $a\oplus 0=0\oplus a=a$	$\exists 1\in 𝒮$, $a\otimes 1=1\otimes a=a$
  $\exists (-a),a+(-a)=0$	if $a\ne 0$, $\exists a^{-1}$, $a\otimes a^{-1}=1$
  $a\oplus b=b\oplus a$	$a\otimes b=b\otimes a$

  Distributivity. $a\otimes (b\oplus c)=a\otimes .b\oplus a\otimes c)$

• The real numbers with addition and multiplication form a field $(ℝ,+,\cdot )$

• The complex numbers with addition and multiplication form a field $(ℂ,+,\cdot )$

• A function $f$ is a relation between two sets $𝒟,𝒞$ that associates to every element of $𝒟$ (the domain) a single element from $𝒞$ (the codomain)

• Notations: $f:𝒟\to 𝒞$, $𝒟\stackrel{f}{\to }𝒞$, $x\in 𝒟$, $f\left(x\right)\in 𝒞$

• An important observation is that vectors can be interpreted as functions:

  • Consider $𝒙\in {ℝ}^n$, $𝒙={\left(\begin{array}{lll}{x}_1& \dots & x_n\end{array}\right)}^T$ (here the ${}^T$ superscript indicates transposition of the row vector into a column vector).

  • Now consider the function $x:\{1,\dots ,n\}\to ℝ$ defined as $x\left(i\right)=x_i$

  The above observations presages that many of the results for the algebra of vectors within ${ℝ}^n$ have relevance and generalization for other types of vectors. In particular, functions $y:ℝ\to ℝ$ can be interpreted as vectors within some set $𝒞$ (different from ${ℝ}^n$)