Friday, 27 September 2013

CSIR NET /SET/ NBHM / GATE / TRB Mathematics: linear algebra

CSIR NET /SET/ NBHM / GATE / TRB Mathematics: linear algebra

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.

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