Overview

• Functions

• Measurements

• Linear mapping composition

Definition. (Relation) . A relation $R$ between two sets $X,Y$ is a subset of the Cartesian product $X\times Y$, $R\subseteq X\times Y$.

Homogeneous relations $H\subseteq A\times A$ are classified according to the following criteria.

Reflection

Relation $H$ is reflexive if $(a,a)\in H$ for any $a\in A$. The equality relation $E\subseteq ℝ\times ℝ$ is reflexive, $\forall a\in A,a=a$, the less than relation $L\subseteq ℝ\times ℝ$ is not, $1\in R,1\nless 1$.

Symmetry

Relation $H$ is symmetric if $(a,b)\in H$ implies that $(b,a)\in H$, $(a,b)\in H\Rightarrow (b,a)\in H$. The equality relation $E\subseteq ℝ\times ℝ$ is symmetric, $a=b\Rightarrow b=a$, the less than relation $L\subseteq ℝ\times ℝ$ is not, $a<b\nRightarrow b<a$.

Anti-symmetry

Relation $H$ is anti-symmetric if $(a,b)\in H$ for $a\ne b$, then $(b,a)\notin H$. The less than relation $L\subseteq ℝ\times ℝ$ is antisymmetric, $a<b\Rightarrow b\nless a$.

Transitivity

Relation $H$ is transitive if $(a,b)\in H$ and $(b,c)\in H$ implies $(a,c)\in H$. for any $a\in A$. The equality relation $E\subseteq ℝ\times ℝ$ is transitive, $a=b\wedge b=c\Rightarrow a=c$, as is the less than relation $L\subseteq ℝ\times ℝ$, $a<b\wedge b<c\Rightarrow a<c$.

Definition. (Function) . A function from set $X$ to set $Y$ is a relation $F\subseteq X\times Y$, that associates to $x\in X$ a single $y\in Y$.

Definition. (Partition) . A partition of a set is a grouping of its elements into non-empty subsets such that every element is included in exactly one subset.

• Equivalence relations partition a set into equivalence classes

Definition. (Norm) . A norm on the vector space $𝒱=(V,S,+,\cdot )$ is a function $||||:V\to {ℝ}_{+}$ that for $𝒖,𝒗\in V$, $a\in S$ satisfies:

1. $||𝒗||=0\Rightarrow 𝒗=𝟎$;

2. $||a𝒖||=\left|a\right|||𝒖||$;

3. $||𝒖+𝒗||⩽||𝒖||+||𝒗||$.

Definition. (Inner Product) . An inner product in the vector space $𝒱=(V,ℝ,+,\cdot )$ is a function $s:V\times V\to ℝ$ with properties

Symmetry

For any $𝒂,𝒙\in V$, $s(𝒂,𝒙)=s(𝒙,𝒂)$.

Linearity in second argument

For any $𝒂,𝒙,𝒚\in V$, $\alpha ,\beta \in ℝ$, $s(𝒂,\alpha 𝒙+\beta 𝒚)=\alpha s(𝒂,𝒙)+\beta s(𝒂,𝒚)$.

Positive definiteness

For any $𝒙\in V\backslash \left\{𝟎\right\}$, $s(𝒙,𝒙)>0$.

• $f:A\to B$ and $g:B\to C$, a composite function, $h=g\circ f$, $h:A\to C$ is defined by

  ${\displaystyle h\left(x\right)=g\left(f\left(x\right)\right).}$

• $𝒇\left(𝒙\right)=𝑨𝒙,𝒈\left(𝒚\right)=𝑩𝒚,𝑨\in {ℝ}^{m\times n},𝑩\in {ℝ}^{p\times m}$

  ${\displaystyle 𝒉\left(𝒙\right)=𝑪𝒙=𝑩𝑨𝒙.}$

  (1)

• Matrix-matrix product is a grouping of matrix-vector products

  ${\displaystyle 𝑪=\left[\begin{array}{cccc}{𝒄}_1& {𝒄}_2& \dots & {𝒄}_n\end{array}\right]=\left[𝑩\begin{array}{cccc}{𝒂}_1& 𝑩{𝒂}_2& \dots & 𝑩{𝒂}_n\end{array}\right]}$