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,v∈V, then u+v∈V.
• SC Scalar Closure
  If α∈C and u∈V, then αu∈V.
• C Commutativity
  If u,v∈V, then u+v=v+u.
• AA Additive Associativity
  If u,v,w∈V, 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 u∈V.
• AI Additive Inverses
  If u∈V, then there exists a vector −u∈V so that u+(−u)=0.
• SMA Scalar Multiplication Associativity
  If α,β∈C and u∈V, then α(βu)=(αβ)u.
• DVA Distributivity across Vector Addition
  If α∈C and u,v∈V, then α(u+v)=αu+αv.
• DSA Distributivity across Scalar Addition
  If α,β∈C and u∈V, then (α+β)u=αu+βu.
• O One
  If u∈V, 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 Date	Subject  (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	A   B  C A   B  C A   B  C A   B  C A   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

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