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.

Joint CSIR UGC NET Exam Question Papers and Keys June 2013

Official website: csirhrdg.res.in

Joint CSIR UGC NET Exam Question Papers
Exam DateSubject  (Booklet Code)Answer Keys  (Booklet Code)
June 2013
Chemical Sciences   A  B  C

Earth Sciences   A  B  C

Life Sciences   A  B  C

Mathematical Sciences   A  B  C

Physical Sciences   A  B   B(E)  C   C(E)

Engineering Sciences   A  B  C
 
  B  C

  B  C

  B  C

  B  C

  B  B(E)  C  C(E)

A  B  C
Dec 2012
Chemical Sciences 

Earth Sciences 

Life Sciences 

Mathematical Sciences 

Physical Sciences 

Engineering Sciences 
 
Key 

Key 

Key 

Key 

Key 

Key  
June 2012
Chemical Sciences 

Earth Sciences 

Life Sciences 

Mathematical Sciences 

Physical Sciences 
 
Key 

Key 

Key 

Key 

Key 
 
Note1: Please note that all paper are in Acrobat PDF format require Acrobat Reader to open. 

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