Vector Spaces 

DefinitionLet $F$ be a field whose elements are scalars and $V$ be a set containing vectors.
A set $V$ is said to be a vector space over a field $F$ if $V$ is closed under finite addition (denoted by +) and scalar multiplication (denoted by $\cdot$). In order of $V$ to become a vector space, the following conditions much hold for all the vectors $u, v, w, x$ $\in$ $V$ and scalars $m, n$ $\in F$: 1) $\vec{u} + \vec{v} = \vec{v} + \vec{u}$ 2) $\vec{u} + (\vec{v} +\vec{w}) = (\vec{u} + \vec{v}) + \vec{w}$ 3) $\vec{0} + \vec{u} = \vec{u} + \vec{0} = \vec{u}$ 4) $\vec{u} + (\vec{u}) = \vec{0}$ 5) $m(n\vec{v}) = (mn)\vec{v}$ 6) $(m + n)\vec{v} = m\vec{v} + n\vec{v}$ 7) $m(\vec{u} + \vec{v}) = m\vec{u} + m\vec{v}$ 8) $1\cdot\vec{v} = \vec{v}$ Note: The above conditions (1  4) to be satisfied by a vector space are under the addition operation and the remaining conditions (5  8) are under the scalar multiplication.
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MotivationThe 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, ntuples, 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.
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Figure 1: 3D Coordinate system representing the game world space; Source: https://clara.io/view/378aae5679d3404c9c5eb1432d916ade/image 
Bird's Eye ViewIf 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. 0 
Context of the DefinitionA 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. Therefore, the two operations "+" and "×" (scalar multiplication) are important and powerful for representing the vectors.
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Figure 2: Addition and scalar mutiplication of vectors 
We call it a 2dimensional 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 3dimensional space because it has three coordinates to represent each vector. 0 
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Figure 3: 3Dimensional Vector Space 
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. $\vec{v} = (v_1, v_2, v_3, . . . . . . . . . . . v_9)$ Like before we can scale these vectors and add them and we can call this universe as a 9Dimensional vector space. $\vec{v} + \vec{u} = (v_1 + u_1, v_2 + u_2, v_3 + u_3, . . . . . . . . . . . v_9 + u_9)$ All the other axioms listed in the definition of a vector space are common for this space as well.

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Figure 4: Addition and scalar multiplication of the vectors as functions 
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.
Note: This example seems trivial but it is of the utmost importance in field theory. 1 
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 nonexamples of vector spaces. Example 1: If we define a set of nonzero real functions $\{ f : \mathbb{R} \rightarrow \mathbb{R}$  $f(x) \neq$ 0 for any $x \in \mathbb{R} \}$, $\implies \begin{bmatrix} 2 \\ 2 \end{bmatrix} \in$ $K$ 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
One vector can be written as the linear combination of the other two. For example, in the second illustration, The formal definition of Linear Combination says that: If $V$ is a vector space over a field $F$ and $V$ = {$v_1$, $v_2$, $v_3$, . . . . . .$v_n$} then $\exists$ a vector $M$ which will be the linear combination of these vectors expressed as $M$ = $(k_1v_1 + k_2v_2 + k_3v_3 + . . . . . . + k_nv_n)$, where $k_1, k_2, . . . . . . . . k_n$ $\in F$.
But why do we call it a 'Linear' Combination? 1 
ApplicationsLinear 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:
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HistoryThe 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. 1 
Pause and Ponder1) Can Riemann integrable functions form a vector space? 1 
References [1] Roman, S., Axler, S., & Gehring, F. W. (2005). Advanced linear algebra (Vol. 3). New York: Springer. 1 
Further Reading [1] https://www.sciencedirect.com/topics/mathematics/linearcombination 1 
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