User:Zackarycw/Sandbox

A vector is an element of a vector space. That is, a vector is an object that it is able to add to other vectors in a commutative and associative manner, is part of a set with an additive identity vector and multiplicative identity scaler, can always find an additive inverse to obtain the additive identity vector, and is able to be scaled by numbers in a commutative and associative manner whose scalar multiplication operation is compatible with field multiplication. Vectors are a specific kind of the more general tensor.

Vector space

A vector space is any object $V$ that is a set over a field $\mathbb {F}$ in which there is vector addition defined, scalar multiplication defined, and meets the eight axioms of a vector space. Vector addition $+:V\times V\to V$, takes two elements in $V$ and assigns them to another another element in $V$. Scalar multiplication $\cdot :\mathbb {F} \times V\to V$, takes an arbitrary number in $\mathbb {F}$ and element of $V$ then assigns them to another element of $V$. The eight axioms of a vector space are:

1. There exists an additive identity element in $V$; $\exists I\in V:\forall v\in V,v+I=I+v=v$
2. There exists a multiplicative identity scalar in $\mathbb {F}$; $\exists J\in \mathbb {F} :\forall v\in V,J\cdot v=v\cdot J=v$
3. There exists an additive inverse for every element in $V$; $\forall v\in V,\exists v^{-1}:v+v^{-1}=v^{-1}+v=I$
4. The vector addition operation is commutative; $\forall u,v\in V,u+v=v+u$
5. The vector addition operation is associative; $\forall u,v,w\in V,\left(u+v\right)+w=u+\left(v+w\right)$
6. The scalar multiplication operation $\cdot$ and field multiplication in $\mathbb {F}$ are compatible; $\forall v\in V\forall a,b\in \mathbb {F} ,a\cdot \left(b\cdot v\right)=\left(ab\right)\cdot v$
7. Scalar multiplication $\cdot$ is distributive over vector addition $+$; $\forall u,v\in V\forall a\in \mathbb {F} ,a\left(u+v\right)=au+av$
8. Scalar multiplication $\cdot$ is distributive over field addition in $\mathbb {F}$; $\forall v\in V\forall a,b\in \mathbb {F} ,\left(a+b\right)v=av+bv$

Elements in $V$ are referred to as vectors and elements in $\mathbb {F}$ are typically called scalars.

Vector subspace

An object $W$ is a vector subspace of a vector space $V$ over field $\mathbb {F}$ if under the operations of $V$, $W$ is a vector space over $\mathbb {F}$. Equivalently, $W$ is a vector subspace of the vector space $V$ over field $\mathbb {F}$ if for every $w_{1},w_{2}\in W$ and $\alpha ,\beta \in \mathbb {F}$ implies that $\alpha w_{1}+\beta w_{2}\in W$. A subspace is nonempty, contains the zero vector, and is closed under vector addition and scalar multiplication. Every vector space is a vector subspace of itself. This is useful as it is sometimes a lot faster to show that $V$ is a vector space via showing that it is a vector subspace of itself rather than showing the entire eight axioms of a vector space. This works as every vector subspace is clearly itself a vector space. If $W$ is a vector subspace of $V$, some authors denote this as $W\leqslant V$.

Dual vector space

Given a vector space $V$ over field $\mathbb {F}$, the dual vector space of $V$, denoted $V^{*}$, is the collection of all linear mappings $\varphi :V\to \mathbb {F}$.

The empty set

The empty set, $\emptyset$, is not a vector space as, since it contains no elements, it doesn't contain an additive identity element or multiplicative identity element. Thus, by definition of a vector, in order for an object to be a vector, the object has to exist ("nothingness" is not a vector).