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Mathematics-2014: Answer Writing Challenge – 2

ARCHIVES

12 September 2014

1) Let V be the vector space of all functions from R into R; let  Ve be the subset of even functions ,

f(–x)=f(x); let Vo be the subset of odd functions, f(–x)=–f(x)

  • Prove that Ve and Vo are subspaces of V.
  • Prove that Ve+Vo=V.
  • Prove that Ve ∩ V0= {0}.

 

2) In V3(R),where R is the field of real numbers, examine  the following  sets of vectors for linear independence:

  • {(1,3,2),(1,-7,-8),(2,1,-1)};
  • {(1,2,0),(0,3,1),(-1,0,1)}.

 

3) Find a linearly independent subset T of the set  S={α1,α2,α3,α4} where α1=(1,2,-1),α2=(-3,-6,3), α3=(2,1,3),α4=(8,7,7)єR3  which spans the same space as S.

 

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