## Friday, 27 September 2013

### linear algebra

##### Vector Space
Definition
Suppose that

V is a set upon which we have defined two operations: (1) 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.
. Then V, along with the two operations, is a vector space over C if the following ten properties hold.
• AC Additive Closure
If u,vV, then u+vV.
• SC Scalar Closure
If αC and uV, then αuV.
• C Commutativity
If u,vV, then u+v=v+u.
• AA Additive Associativity
If u,v,wV, then u+(v+w)=(u+v)+w.
• Z Zero Vector
There is a vector, 0, called the zero vector, such that u+0=u for all uV.
• AI Additive Inverses
If uV, then there exists a vector uV so that u+(u)=0.
• SMA Scalar Multiplication Associativity
If α,βC and uV, then α(βu)=(αβ)u.
• DVA Distributivity across Vector Addition
If αC and u,vV, then α(u+v)=αu+αv.
• DSA Distributivity across Scalar Addition
If α,βC and uV, then (α+β)u=αu+βu.
• O One
If uV, then 1u=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.