Difference between revisions of "Vector space"

Revision as of 07:37, 23 August 2015 (view source) JessicaBelinda133 (Talk | contribs) ← Older edit		Latest revision as of 16:30, 23 August 2015 (view source) Alec (Talk | contribs) m (Reverted edits by JessicaBelinda133 (talk) to last revision by Alec)
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−	An introduction to the important concepts of vector spades and linear algebra may be found on the [[Basis and coordinates]] page	+	An introduction to the important concepts of vector spaces and linear algebra may be found on the [[Basis and coordinates]] page
	==Definition==		==Definition==
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	We denote a vector space as "Let <math>(V,F)</math> be a vector space" often we will write simply "let <math>V</math> be a vector space" if it is understood what the field is, because [[Mathematicians are lazy|mathematicians are lazy]]		We denote a vector space as "Let <math>(V,F)</math> be a vector space" often we will write simply "let <math>V</math> be a vector space" if it is understood what the field is, because [[Mathematicians are lazy|mathematicians are lazy]]
−	A [[Norm|normed]] vector space may be demoted to <math>(V,\|\cdot\|_V,F)</math>	+	A [[Norm|normed]] vector space may be denoted <math>(V,\|\cdot\|_V,F)</math>
	===Example===		===Example===
	Take <math>\mathbb{R}^n</math>, an entry <math>v\in\mathbb{R}^n</math> may be denoted <math>(v_1,...,v_n)=v</math>, scalar multiplication and addition are defined as follows:		Take <math>\mathbb{R}^n</math>, an entry <math>v\in\mathbb{R}^n</math> may be denoted <math>(v_1,...,v_n)=v</math>, scalar multiplication and addition are defined as follows:
	* <math>\lambda\in\mathbb{R},v\in\mathbb{R}^n</math> we define scalar multiplication <math>\lambda v=(\lambda v_1,...,\lambda v_n)</math>		* <math>\lambda\in\mathbb{R},v\in\mathbb{R}^n</math> we define scalar multiplication <math>\lambda v=(\lambda v_1,...,\lambda v_n)</math>
−	* <math>u,v\in\mathbb{R}^n</math> - we define addition as <math>u+v=(u_1+v_1,...,u_n+v_n)</math> so like add the parts	+	* <math>u,v\in\mathbb{R}^n</math> - we define addition as <math>u+v=(u_1+v_1,...,u_n+v_n)</math>
	==Important concepts==		==Important concepts==

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Latest revision as of 16:30, 23 August 2015

An introduction to the important concepts of vector spaces and linear algebra may be found on the Basis and coordinates page

Definition

A vector space $V$ over a field $F$ is a non empty set $V$ and the binary operations:

• $$+:V\times V\rightarrow V$$ given by $$+(x,y)=x+y$$ - vector addition
• $$\times:F\times V\rightarrow V$$ given by $$\times(\lambda,x)=\lambda x$$ - scalar multiplication

Such that the following 8 "axioms of a vector space" hold

Axioms of a vector space

1. $$(x+y)+z=x+(y+z)\ \forall x,y,z\in V$$
2. $$x+y=y+x\ \forall x,y\in V$$
3. $$\exists e_a\in V\forall x\in V:x+e_a=x$$ - this $$e_a$$ is denoted $$0$$ once proved unique.
4. $$\forall x\in V\ \exists y\in V:x+y=e_a$$ - this $$y$$ is denoted $$-x$$ once proved unique.
5. $$\lambda(x+y)=\lambda x+\lambda y\ \forall\lambda\in F,\ x,y\in V$$
6. $$(\lambda+\mu)x = \lambda x+\mu x\ \forall\lambda,\mu\in F,\ x\in V$$
7. $$\lambda(\mu x)=(\lambda\mu)x\ \forall\lambda,\mu\in F,\ x\in V$$
8. $$\exists e_m\in F\forall x\in V:e_m x = x$$ - this $$e_m$$ is denoted $$1$$ once proved unique.

Notation

We denote a vector space as "Let

$$(V,F)$$ be a vector space" often we will write simply "let $$V$$ be a vector space" if it is understood what the field is, because mathematicians are lazy

A normed vector space may be denoted

$$(V,\|\cdot\|_V,F)$$

Example

Take

$$\mathbb{R}^n$$ , an entry $$v\in\mathbb{R}^n$$ may be denoted $$(v_1,...,v_n)=v$$ , scalar multiplication and addition are defined as follows:

• $$\lambda\in\mathbb{R},v\in\mathbb{R}^n$$ we define scalar multiplication $$\lambda v=(\lambda v_1,...,\lambda v_n)$$
• $$u,v\in\mathbb{R}^n$$ - we define addition as $$u+v=(u_1+v_1,...,u_n+v_n)$$

Important concepts

• Linear maps - the homomorphisms and isomorphisms of vector spaces
• Span, linear independence, linear dependence, basis and dimension
• Norm
• Linear isometry