Let V be the vector space of all functions from R to R Descr

Let V be the vector space of all functions from R to R. Describe, in a simple and familiar way, the subspace of V spanned by the functions f(x) = (n + x)^n for all positive integers n 1

Solution

Given function:f(x) = (n + x)^n for n1

                Suppose that v 1v2,n were linearly dependent. Then n >1,

and there would exist scalars a1 , a 2, . , an, not all zero, such that a1 v1+a2v2+··   +anvn=0.

We may suppose, without loss of generality, that an 6= 0. Then vn=b1v1+b2v2+··+bn1vn1,

where bi=aia1 and i= 1,2, . . . , n1.

But thenv1,v2, . . . ,vn1would span V, since any linear combination of v1,v2, . . . ,vn

could be expressed as a linear combination of v1,v2, . . . ,vn1.

But the definition of n ensures that no set of n- 1 elements of S can span V.

We conclude that v1,v2,v3-----vn must be linearly independent, and thus must constitute a basis of V as required.

i,e (n+x)^n=0 n 1 for case(1)

     and   vn=b1v1+b2v2+------bnvn.   where bn sone function as per our problem