Let (\(F,+,\cdot)\) be a field. A non-empty set \(V\) of elements called **vectors** with two maps.
\[
\cases{
V\times V\mapsto V:(x,y)\mapsto x\oplus y & \text{(the vector addition)}\\
F\times V\mapsto V:(\alpha,x)\mapsto \alpha \odot x & \text{(the scalar multiplication)}
}\]

is called a **vector space over the field** $F$ (or a *$F$-vector space* or a **linear space over** $F$), if:
1. \((V,\oplus)\) is an Abelian group, in particular:
* The addition "$\oplus$" of vectors is associative: $x\oplus (y\oplus z)=(x\oplus y)\oplus z$ for all $x,y,z\in V.$
* "$\oplus$" has a neutral element, called the **zero vector** $o\in V$ with $x\oplus o=o\oplus x=x$ for all $x\in V.$
* Every vector $x\in V$ has an inverse $-x\in V$ with $x\oplus(-x)=(-x)\oplus x=o.$
* "$\oplus$" is commutative, i.e. $x\oplus y=y\oplus x$ for all $x,y\in V.$
1. For \(\alpha,\beta\in F\) and \(x,y\in V\) the following **axioms of scalar multiplication** hold:
* $(\alpha + \beta)\odot x=\alpha\odot x \oplus \beta\odot x,$
* $\alpha\odot(x\oplus y)=\alpha\odot x \oplus \alpha\odot y,$
* $(\alpha\cdot \beta)\odot x=\alpha\odot (\beta\odot x),$
* If $1\in F$ is the neutral element of the field, then it is also a neutral element of the scalar multiplication in $V,$ i.e. $1\odot x=x.$

Whenever any misunderstandings can be excluded, **BookofProofs** follows the convention used in most mathematical literature, making no difference in the notation of vector addition "$\oplus$" and the addition of field elements "$+$", as well as the scalar multiplication "$\odot$" and the multiplication of field elements "$\cdot$".
However, to keep field elements and vectors apart, the latter will be denoted by small Latin letters, e.g. $x,y,a,b,\ldots,$ while field elements will be denoted by small Greek letters $\alpha,\beta,\gamma,\ldots.$

If \(V\) has a finite basis, \(B=\{b_1,\ldots, b_n\}\subseteq V\), the uniqueness lemma ensures that we can write any vector \(v\in V\) as a unique linear combination of the basis vectors: \[v = \alpha_1b_1+\ldots+\alpha_nb_n.\] With respect to the basis \(B\), we can write the vector \(v\) in its column notation

\[ v=\pmatrix{ \alpha_{1}\cr \alpha_{2}\cr \vdots \cr \alpha_{n} \cr } \]

There are some things to mention:

- This notation depends on the chosen basis, i.e. the same vector will have different notations for two different bases.
- Writing the vector as a column is a common convention.
- The above-mentioned rules of \((V, \oplus)\) being an Abelian group as well as scalar multiplication \(\odot\) are easily verified for the following operations (for \(\alpha_i,\beta_i,\lambda\in F\)):

\[\text{vector addition }\oplus:=\pmatrix{ \alpha_{1}\cr \alpha_{2}\cr \vdots \cr \alpha_{n} \cr }+\pmatrix{ \beta_{1}\cr \beta_{2}\cr \vdots \cr \beta_{n} \cr }=\pmatrix{ \alpha_{1}+\beta_1\cr \alpha_{2}+\beta_2\cr \vdots \cr \alpha_{n}+\beta_n \cr } \] \[\text{scalar multiplication}\odot:=\lambda\cdot\pmatrix{ \alpha_{1}\cr \alpha_{2}\cr \vdots \cr \alpha_{n} \cr }=\pmatrix{ \lambda\cdot\alpha_{1}\cr \lambda\cdot\alpha_{2}\cr \vdots \cr \lambda\cdot\alpha_{n} \cr } \]

Examples: 1

Applications: 1

Chapters: 2 3 4 5

Corollaries: 6 7 8

Definitions: 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Examples: 43 44 45

Explanations: 46

Lemmas: 47

Motivations: 48

Parts: 49 50

Proofs: 51 52 53 54 55 56 57 58 59

Propositions: 60 61 62 63 64 65 66 67 68 69 70