Definition

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Motivation

The concept of vector spaces came long back in 1888. It took decades to comprehend the idea and realize its significance. At first, vectors were dealt with many approaches. They were treated like arrows, n-tuples, list of numbers etc and the space containing those vectors were called vector spaces. But later the notion of vector space was generalized to be a space containing not only the elements like tuples or arrows, but it can be any space where the addition and scalar multiplication of its elements are allowed.
You must have noticed that most of the video games are defined in a 3-dimensional world and the characters of the game can be moved in whichever position you want. If the character in the game is moving 4 units forward, 2 units upwards, then just because video games use numbers to represent a character's position on the screen, the video game screen will function like a 3D coordinate plane. So, as the position of the character changes in the game, the coordinates in 3D plane changes too. If the initial position of the character is (4, 2, 2) and now, if it is moving 5 units forward, then the two coordinates will get added and will be (9, 2, 2).
In this case, all the elements of 3D coordinate space can be treated like vectors and those vectors form a vector space.

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Bird's Eye View

If you ask someone to define a vector space, you get the definition in terms of vectors, and then if you ask to define a vector, you get the definition in terms of a vector space. Now, this is a cycle that we'll not break since both can be very well explained in terms of each other.
A vector space can be described as a space where all the "similar" vectors live. But what do you mean by that? Considering the previous example of a 3-dimensional space in a video game where the vectors were in the form (x, y, z), we see that the space is populated by all the similar kinds of vectors with 3 coordinates.
Similarly, if the space were 2-dimensional, then all the vectors were of the same kind and the form (x, y).
A real space is a vector space in which all the elements are real numbers. Now in addition to the similarity criteria, the vectors must also be closed under addition and scalar multiplication to form a vector space. Therefore we can think of a vector space as a space of similar vectors that can be added and multiplied by any scalar.

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Context of the Definition

A vector space, in simpler words, is an algebraic structure where "addition" and "scalar multiplication" of the inhabitant vectors are allowed and in fact, are the only allowed operations.
In the plane $\mathbb{R}^2$, a vector can be described as a unique pair of coordinates (x, y). If any two vectors are added, it gives a third vector with a different pair of coordinates. And again, those vectors can also be scaled by any quantity. We can choose our scalars to be anything like a real number or it can be a complex number as well. Based on that, we can call that space as "real" vector space or a "complex" vector space. But we often prefer real numbers as scalars.

Therefore, the two operations "+" and "×" (scalar multiplication) are important and powerful for representing the vectors.

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We call it a 2-dimensional vector space because it takes two coordinates as (x, y). Likewise, we can choose vectors with three coordinates namely x, y, and z. We can add and scale the vectors in a plane $\mathbb{R}^3$ as well and get a third vector. These vectors form a 3-dimensional space because it has three coordinates to represent each vector.
 

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So the definition of the vector space holds for $\mathbb{R}^1$, $\mathbb{R}^2$ and $\mathbb{R}^3$. But it would be pretty much difficult and ambiguous to show the same thing geometrically if we consider a 4- or more dimensional space. 
In Abstract Algebra certain things are beyond our imagination and we aim to generalize them as much as possible. 
Imagine living in a parallel universe with 9-dimensions. Intuitively, it would be correct if we assume that this universe has a space which is a collection of 9 forces (vectors), an inspiration from 2- and 3- dimensional spaces. 

$\vec{v} = (v_1, v_2, v_3, . . . . . . . . . . . v_9)$
$\vec{u} = (u_1, u_2, u_3, . . . . . . . . . . . u_9)$

Like before we can scale these vectors and add them and we can call this universe as a 9-Dimensional vector space.

$\vec{v} + \vec{u} = (v_1 + u_1, v_2 + u_2, v_3 + u_3, . . . . . . . . . . . v_9 + u_9)$
c$\cdot\vec{v} = (cv_1, cv_2, cv_3, . . . . . . . . . cv_9)$

All the other axioms listed in the definition of a vector space are common for this space as well.
Now, we can generalize a vector space to be a set of multi-dimensional vectors, together with a set of 1-Dimensional quantities called scalars. Moreover, we can say that the listed axioms are also true for any n-Dimensional space having the vectors with n-coordinates. We can call this as Euclidean space or n-space. 
Vector spaces are basic to Linear Algebra and appear throughout the Mathematics.

But...Are there any other vector spaces? 

So far we have considered a vector space to be a collection of vectors that are simply an array of real numbers or coordinates. But that would be unfair to the other spaces if we fundamentally visualize the vectors to be just an "arrow" or an "array" of real numbers because they are something more than these typical vectors. They too have all the vector-like qualities and that follows all the axioms of vector spaces. 
So, let's start with Functions! That does make sense that adding any two real-valued functions will give another function and the same works for scaling them by any scalar. For example, Let there be any two functions $f(x) = sin(x)$ and $g(x) = cos(x)$ and now if we add or scale them, we get another function.
$\implies f(x) + g(x) = sin(x) + cos(x)$ and $kf(x) = ksin(x)$ where k is any scalar.  Therefore, functions can be treated like vectors in a space.

It can be formally defined as follows:
Let $V$ be a vector space over a field $F$ and $A$ be any set. If we define functions $f : A \rightarrow V$ and $g : A \rightarrow V$ then for any $x \in A$, $c \in F$

•  $(f + g)(x) = f(x) + g(x)$
• $f(cx) = cf(x)$

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Here are some other examples of the vector spaces:

Example 1: The set of $M_2(\mathbb{Q})$ of 2 x 2 matrices with rational entries is a vector space over the rational field.

                   $\begin{bmatrix} u_1 & u_2 \\ u_3 & u_4 \end{bmatrix} +  \begin{bmatrix} v_1 & v_2 \\ v_3 & v_4 \end{bmatrix} = \begin{bmatrix} u_1 + v_1  & u_2 + v_2 \\ u_3 + v_3 &  u_4 + v_4 \end{bmatrix}$  and

                  $c \begin{bmatrix} u_1 & u_2 \\ u_3 & u_4 \end{bmatrix}$ = $\begin{bmatrix} cu_1 & cu_2 \\ cu_3 & cu_4 \end{bmatrix}$ $\forall c \in \mathbb{Q}$ and

                 Also, $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ $\in M(\mathbb{Q})$

                 Hence, it is a vector space.

Example 2: Let $P(\mathbb{R})$ be a set of all the polynomials with the coefficients from the field $\mathbb{R}$.
                    In $P(\mathbb{R})$, the two elements $P_1 = (a_0 + a_1x  + a_2x^2 + a_3x^3 + . . . . . . . . . .+a_nx^n)$ and
                    $P_2 = (b_0 + b_1x + b_2x^2 + . . . . . . . . b_nx^n)$ of degree n where $a_1, a_2, . . . . . . .+ a_n \in \mathbb{R}$ and 
                    $b_1, b_2, b_3, . . . . . .+  b_n  \in \mathbb{R}, a_n \neq$ 0 and $b_n \neq$ 0
                    $\implies$  0 + 0$x$ + 0$x^2$ + . . . . . . .+ 0$x^n \in$ $P(\mathbb{R})$
                    $\implies$ $P_1$ + $P_2$ = $(a_0 + b_0) + (a_1 + b_1)x  + (a_2 + b_2)x^2 + (a_3 + b_3)x^3 + . . . . . . . . . .+ (a_n + b_n)x^n \in$ $P(\mathbb{R})$ and 
                    $\implies$ $cP_1$ = $ca_0 + ca_1x  + ca_2x^2 + ca_3x^3 + . . . . . . . . . .+ ca_nx^n \in$ $P(\mathbb{R})$
                    Hence, it is a vector space.

Example 3: The set of complex numbers $C = \{ a + bi$ | $a, b \in \mathbb{R} \}$ forms a vector space over $\mathbb{R}$.
                     The vector addition and scalar multiplication are usual addition and multiplication of complex
                     numbers and  (0 + 0$i) \in$ $C$.

Example 4:  Let $E$ be the field and $F$ be its subfield. Then $E$ is a vector space over $F$. The vector addition and scalar multiplication follows the operations of $E$.

                     Note:  This example seems trivial but it is of the utmost importance in field theory.
                                 For better understanding refer to [2] in further reading.
 

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What isn't a vector space?

Anything that doesn't obey the vector space axioms is not a vector space. Do observe that even if one of the defined axioms is not satisfied then it will not be a vector space. The following are some non-examples of vector spaces.

 Example 1:  If we define a set of non-zero real functions $\{ f : \mathbb{R} \rightarrow \mathbb{R}$ | $f(x) \neq$ 0 for any $x \in \mathbb{R} \}$,
                      then it won't form a vector space.
                      Take $f(x) = x^2$ and another such function $g(x) = -5$ then both $f(x) \neq$ 0 and $g(x) \neq$ 0
                      but their addition $(f + g)(x) = x^2 + 1 - 5 = (x + 2)(x - 2)$ is not non-zero, since $(f+g)(2) = 0$
                      $\implies$ It is not closed under vector addition.
                      Also the additive identity will be a zero function which is not in the set.
                      Hence, the set of non-zero functions is not a vector space.

Example 2:  Let $K$ = $\left\{ \begin{bmatrix} m \\ n \end{bmatrix} | m, n \geq 0 \right\}$

                     $\implies \begin{bmatrix} 2 \\ 2 \end{bmatrix} \in$ $K$

                     But then consider any scalar, say -3 $\in \mathbb{R}$ then

                    -$3  \begin{bmatrix} 2 \\ 2 \end{bmatrix} =  \begin{bmatrix} -6 \\ -6 \end{bmatrix} \notin$ $K$

                     $\implies$ It is not closed under vector addition
                     Hence, it is not a vector space 

By now you must have realized that vectors are vivid and can be added and scaled endlessly. We can expand them by scaling and adding in any way and get another vector which will be called as a 'Linear Combination' of those vectors.

One vector can be written as the linear combination of the other two. For example, in the second illustration,
2$\vec{a}$ + 3$\vec{b}$ = $\vec{c}$ and in fourth, $h(x) = 2f(x)$ + $g(x)$.

The formal definition of Linear Combination says that:

But why do we call it a 'Linear' Combination?
Is it because we are just scaling them up and adding them? 
The answer is a resounding YES! Because here, if we perform any other operation instead of addition or scalar multiplication then it will become a non-linear combination. So all we do is scale and add them. The 'scaling property' is what makes it linear and that's why we call it a 'Linear' Combination.
 

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Applications

Linear Algebra is considered an important concept that has many engineering applications. Physicists, game developers, and data scientists, all have this one thing in common: Vectors! There are numerous obvious and solid applications of vector spaces. Here are some:

• Aircraft Vectoring
  The goal of vectoring is to manage the traffic flow and retain the desired track. Air traffic controllers use vector algebra to recognize distances and directions at a moment's notice and notifies the pilots to move in a particular direction for a specific magnitude.
• Vector Space Search Engines
  Another very good application of vector spaces is building a vector space search engine. Most of the search engines use a vector space model that includes converting documents into vectors in a high dimensional space. The number of dimensions depends on the number of unique words. The document's status is inferred by the words it has.
  The documents with similar words end up close together and the documents that have only a few words in common end up far apart.

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History

The abstract theory of vector space came into the picture very long back in the18th century. But what we have up to now became clear after decades of work... It all began with the discussion of vector spaces $\mathbb{R}^2$ and $\mathbb{R}^3$ by the two french mathematicians Fermat and Descartes, in a similar way that we are studying now. But they intensified more on points and plotting rather than the idea of a vector. Meanwhile, in 1804, Bolzano brought up certain operations on points, lines, and planes which are prototypes of vectors. The generalized and concrete definition of a vector space didn't appear until the late 19th century. But then, the Italian mathematician Giuseppe Peano, in 1888 presented the modern definition and axioms of a vector space. 
He dealt with vectors in different approaches. His first approach, in 1887, was performing addition and scalar multiplication in n-tuples. He didn't call them vectors at that time. After this, he tried a second way and identified it as a line segment. The third approach in 1888 was what he called 'linear systems' which are now known as vector or linear spaces. 
It was Peano's third approach that was important and what we are studying now. The first approach was not suitable and axiomatic, and the second one was formulated by him a decade later. He also defined Linear Maps in the same year. Peano's theory was not approved soon after he proposed it in 1888. The first person to adopt his axioms for linear systems was Cesare Burali-Forti, in 1896. He was Peano's colleague at the military academy in Turin. Later, it was accepted by many other people. By the end of the century, he discovered many important concepts other than vector spaces, that include differentiability, convex sets, limits of families of sets, and tangent cones.
 

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Pause and Ponder

1)  Can Riemann integrable functions form a vector space?
2)  In an RGB color space, any color can be formed by taking the combination of the primary colors: red, green, and blue. Is it a vector space?

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References

 [1] Roman, S., Axler, S., & Gehring, F. W. (2005). Advanced linear algebra (Vol. 3). New York: Springer.
 [2] Gallian, J. (2012). Contemporary abstract algebra. Nelson Education.
 [3] Moore, G. H. (1995). The axiomatization of linear algebra: 1875-1940. Historia Mathematica, 22(3), 262-303.
 [4] Dolecki, S., & Greco, G. H. (2010). Towards historical roots of necessary conditions of optimality. Regula of Peano.
       arXivpreprint arXiv:1002.4581.
 [5] https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron%2C_Cherney%2C_and_Denton)05%3A_Vector_Spaces/5.01%3A_Examples_of_Vector_Spaces
 [6] https://www.perl.com/pub/2003/02/19/engine.html/

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Further Reading

 [1] https://www.sciencedirect.com/topics/mathematics/linear-combination
 [2] https://math.stackexchange.com/questions/1939227/how-to-show-simply-that-a-field-is-a-vector-space-over-any-of-its-subfields

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