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Mathematics-2014: Answer Writing Challenge – 4


17 September 2014


1) Show that the set S={1,x,x2,……,xn} of n+1 polynomials in x is a basis of the vector space Pn(R), of all polynomials in x (of degree at most n) over the field of real numbers.



2) “Corresponding to each subspace W1 of a finite dimensional vector space V(F),there exists a subspace W2 such that V is the direct sum of W1 and W2.”

Prove the theorem.



3) Let V be the vector space of all polynomial functions of degree less than or equal to two from the field of real numbers R into itself. For a fixed  tєR, let  g1(x)=1, g2(x)=x+t, g3(x)=(x+t)2.

Prove that  {g1,g2,g3} is a basis for V and obtain the coordinates of c0+c1x+c2x2.