Rigorous Definition of Scalars and Vectors? [closed]

Question

What is the rigorous definition of "Vector" (& " Scalar")? Best I got was:

https://www.youtube.com/watch?v=Ncx98PmXbZc

Answer 1

The short answer is:

A vector is an element of a vector space, whereas a scalar is an element of the underlying field

In order to learn more about vector spaces you can use any introductory linear algebra book.

Long answer:

Definition Field:

Let $F$ be a set endowed with an interior operation.

$+: F \times F \longrightarrow F, (a, b) \mapsto a+b$

(This operation can be anything. You put in 2 elements of the set and get 1 back).

This operation should fullfill

1. Associativity: i.e. $(a+b)+c=a+(b+c)$

2. Commutativty: i.e $a+b=b+a$

3. There should be a neutral element, i.e. $\exists e \in K: a+e= a \ \ \forall \ \ a \in K $

   (The neutral element is usually denoted by $0$)

4. For each element ther should be an inverse, i.e. $\forall a \in K \exists b \in K \ \ s.t. a+b = 0$

   (The inverse is usually denoted by -a)

This first opertion is called an addition

Furthermore, there is second operation:

$\cdot : F \times F \longrightarrow F: (a, b) \mapsto a\cdot b$

With respect to which $F/0$ (Taking the neutral away) fullfills properties 1-4, i.e.

1. Associativity $(a \cdot b) \cdot c = a \cdot (b \cdot c)$

2. Commutivity $a \cdot b= b \cdot a$

3. Existence of neutral element (denoted by $1$)

4. Existence of inverse for all elements except $0$

Furthermore, there is distributivity between the operations, i.e.

$(a+b)\cdot c = a \cdot c + b \cdot c$

We call the triple $(F, +, \cdot)$ a field

Examples of fields include $\mathbb{R}, \mathbb{C}$

Definition Vector space:

A vector space $V$ over a field $F$ is a set, endowed with an addition

$+ : V \times V \longrightarrow V$

with the same property as the addition in the field

and a scalar mulitplication:

$\cdot: F \times V \longrightarrow V, (a, v) \mapsto a \cdot v,$

with: $1 \cdot v = v, \forall v \in V$

and distributivty, i.e for $a, b \in F,$ $v, w \in V$

$(a + b) \cdot v = a \cdot v + b \cdot v$

$a \cdot (v+w)= a \cdot v+b \cdot w$

Furthermore it respects the associativity of the field multiplication, i.e.

$(a\cdot b) \cdot v = a \cdot (b \cdot v)$ for $a, b \in F,$ $v \in V$

The elements of the field are called scalars the elements of the vector space are called vectors.

This is the formal definition