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.
