The concept of Vector space is not straight forward as it sounds. Unlike the usual ones, it doesn't have a perfect physical basis in reality starting with the question "Why".
But the ideas are not completely abstract as you think, it was created not from a specific topic in physics, but from the generalization of all the usual mathematical concepts.
I will try to go with my own formal introduction, where everything could be started and understood from the beginning of Quantum Mechanics.
When the idea of Wave function and operations of quantum mechanics are introduced, people really don't understand the insights. They just used all the arithmetical manipulations and concepts from the known classical mechanics and applied it into quantum mechanics in terms of operators.
But they never know, why those operators behave in a classical form and why it explains the Nature so beautifully and so on with many philosophical questions.
You may ask, then why people work with a mathematics for which, they themselves don't know the reason "why" it works.
As scientists, they have other things to create and work with in real life instead of simply getting into the philosophical questions.
After all, applications are more important than the complete reasoning.
A single Hydrogen atom in Earth gives a spectrum that is exactly as same as in the Jupiter. We can use this property to communicate, even if we don't know the answer for the question "why exactly it works the same?".
So they just said, "The mathematics works fine. What do we need extra other than that, to apply it in real life!!" .
Eventually, they stopped about thinking "Why" and proceeded to define "How" things can be developed in this new mathematics with the help of introducing some new abstract concepts.
This is the reason why, Quantum Mechanics looks as it has more postulates than any other field in Physics.
From these numerous abstract postulates, they developed a whole lot of other concepts and succeeded with it in real life. And so, Quantum Mechanics was formulated.
As we did go along, we found that these abstract definitions and operations plays crucial role not only in Quantum Mechanics but also in many other places.
From this, it was believed that, may be there is some basic mathematics intrinsically hidden in Nature.
To understand more, accordingly, they combined it together and found out some of the most common basic rules followed by all those abstract quantities and initiated the concept of Vector space.
Linear Vector Space:
[Note that, the following concepts are not the first and newly defined but they are just the compilation of basic concepts you can find it anywhere in physics.]
A vector is a mathematical notation or an entity used to denote a concept. As we used to express the whole of Nature itself using numbers, these concepts are also intrinsically related with one another with the help of numbers.
The numbers can be either real, imaginary or complex, etc. These numbers form a field, i.e. just a new name to denote the set of numbers that is used to related these vectors. If they are scalar numbers, then it is called scalar field.
Let us denote the set of vector elements by V and the set of field elements by F. They should obey some fundamental axioms to be defined as the Vector space.
To be considered as a field, the set F should follow these axioms,
Closure: For all two elements 'a' and 'b' , $ a\,,b\, \in F $ then $ a*b \in F $ where * denotes any binary operation. The most usual one is addition and multiplication.
Associativity: For all three elements $a,\,,b\,,c \in F $ there exists an equality $$ a*(b*c) = (a*b)*c $$
Commutativity: For all two elements $ a,\,b \in F $ there exists an equality, $$ a*b = b*a $$
Existence of Identity: For all elements $ a \in F $ there exists an identity element "e" such that, $$ a * e = a $$
Existence of Inverse: For all element $ a \in F $ there exists an inverse element $ a^{-1} $ such that $$ a * a^{-1} = e $$ where "e" is the identity element.
Distributivity: If two operations are considered i.e. addition and multiplication then distributivity is defined by the condition that for all three elements $ a,\,b\,,c \in F $ there is an equality $$ a(b+c) = ab + ac $$
The operations needn't be addition and subtraction but can be any binary operation.
Once the above axioms are satisfied, the set is called a "field". Now, proceeding to the next set of vector elements "V" it has to satisfy the following axioms similar to the old one, but now we take two vector elements.
Associativity, commutativity, identity and inverse are defined as the same as previous.
The new properties are ,
compatibility with scalar multiplication with a field element. For $ a,b\in F $ and $ \vec{u} \in V $
$$ a(b\vec{u}) = (ab) \vec{u} $$
Distributivity with scalar multiplication with vector addition, For all $ a \in F $ and $ \vec{u}, \vec{v} \in V $ there exists, $$ a(\vec{u} +\vec{v}) = a\vec{u} + a\vec{v} $$
When I denote vector elements with vector notation, it doesn't mean it is the usual three dimensional vector. I am just using it for the notation consistency and nothing more!
But the ideas are not completely abstract as you think, it was created not from a specific topic in physics, but from the generalization of all the usual mathematical concepts.
I will try to go with my own formal introduction, where everything could be started and understood from the beginning of Quantum Mechanics.
When the idea of Wave function and operations of quantum mechanics are introduced, people really don't understand the insights. They just used all the arithmetical manipulations and concepts from the known classical mechanics and applied it into quantum mechanics in terms of operators.
But they never know, why those operators behave in a classical form and why it explains the Nature so beautifully and so on with many philosophical questions.
As scientists, they have other things to create and work with in real life instead of simply getting into the philosophical questions.
After all, applications are more important than the complete reasoning.
A single Hydrogen atom in Earth gives a spectrum that is exactly as same as in the Jupiter. We can use this property to communicate, even if we don't know the answer for the question "why exactly it works the same?".
So they just said, "The mathematics works fine. What do we need extra other than that, to apply it in real life!!" .
This is the reason why, Quantum Mechanics looks as it has more postulates than any other field in Physics.
From these numerous abstract postulates, they developed a whole lot of other concepts and succeeded with it in real life. And so, Quantum Mechanics was formulated.
As we did go along, we found that these abstract definitions and operations plays crucial role not only in Quantum Mechanics but also in many other places.
From this, it was believed that, may be there is some basic mathematics intrinsically hidden in Nature.
To understand more, accordingly, they combined it together and found out some of the most common basic rules followed by all those abstract quantities and initiated the concept of Vector space.
Linear Vector Space:
[Note that, the following concepts are not the first and newly defined but they are just the compilation of basic concepts you can find it anywhere in physics.]
A vector is a mathematical notation or an entity used to denote a concept. As we used to express the whole of Nature itself using numbers, these concepts are also intrinsically related with one another with the help of numbers.
The numbers can be either real, imaginary or complex, etc. These numbers form a field, i.e. just a new name to denote the set of numbers that is used to related these vectors. If they are scalar numbers, then it is called scalar field.
Let us denote the set of vector elements by V and the set of field elements by F. They should obey some fundamental axioms to be defined as the Vector space.
To be considered as a field, the set F should follow these axioms,
Closure: For all two elements 'a' and 'b' , $ a\,,b\, \in F $ then $ a*b \in F $ where * denotes any binary operation. The most usual one is addition and multiplication.
Associativity: For all three elements $a,\,,b\,,c \in F $ there exists an equality $$ a*(b*c) = (a*b)*c $$
Commutativity: For all two elements $ a,\,b \in F $ there exists an equality, $$ a*b = b*a $$
Existence of Identity: For all elements $ a \in F $ there exists an identity element "e" such that, $$ a * e = a $$
Existence of Inverse: For all element $ a \in F $ there exists an inverse element $ a^{-1} $ such that $$ a * a^{-1} = e $$ where "e" is the identity element.
Distributivity: If two operations are considered i.e. addition and multiplication then distributivity is defined by the condition that for all three elements $ a,\,b\,,c \in F $ there is an equality $$ a(b+c) = ab + ac $$
The operations needn't be addition and subtraction but can be any binary operation.
Once the above axioms are satisfied, the set is called a "field". Now, proceeding to the next set of vector elements "V" it has to satisfy the following axioms similar to the old one, but now we take two vector elements.
Associativity, commutativity, identity and inverse are defined as the same as previous.
The new properties are ,
compatibility with scalar multiplication with a field element. For $ a,b\in F $ and $ \vec{u} \in V $
$$ a(b\vec{u}) = (ab) \vec{u} $$
Distributivity with scalar multiplication with vector addition, For all $ a \in F $ and $ \vec{u}, \vec{v} \in V $ there exists, $$ a(\vec{u} +\vec{v}) = a\vec{u} + a\vec{v} $$
When I denote vector elements with vector notation, it doesn't mean it is the usual three dimensional vector. I am just using it for the notation consistency and nothing more!
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