## Friday, 27 September 2013

### linear algebra

##### Vector Space

Definition

Suppose that

*vector addition*, which combines two elements of V and is denoted by “+”, and (2)

*scalar multiplication*, which combines a complex number with an element of V.

*vector space*over

- AC Additive Closure

Ifu,v∈V , thenu+v∈V . - SC Scalar Closure

Ifα∈C andu∈V , thenαu∈V . - C Commutativity

Ifu,v∈V , thenu+v=v+u . - AA Additive Associativity

Ifu,v,w∈V , thenu+(v+w)=(u+v)+w . - Z Zero Vector

There is a vector,0 , called the zero vector, such thatu+0=u for allu∈V . - AI Additive Inverses

Ifu∈V , then there exists a vector−u∈V so thatu+(−u)=0 . - SMA Scalar Multiplication Associativity

Ifα,β∈C andu∈V , thenα(βu)=(αβ)u . - DVA Distributivity across Vector Addition

Ifα∈C andu,v∈V , thenα(u+v)=αu+αv . - DSA Distributivity across Scalar Addition

Ifα,β∈C andu∈V , then(α+β)u=αu+βu . - O One

Ifu∈V , then1u=u .

The objects in V are called

*vectors*, no matter what else they might really be, simply by virtue of being elements of a vector space.
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