Definition: Vector Space

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. $1\odot x=x.$

A note on notation

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 $\alpha,\beta,\gamma,\ldots.$

Vector notation with respect to a basis

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:

  1. This notation depends on the chosen basis, i.e. the same vector will have different notations for two different bases.
  2. Writing the vector as a column is a common convention.
  3. 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

  1. Proposition: Simple Consequences from the Definition of a Vector Space

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

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