Let Hom(V,W) be the set of linear transformations from V to W. Define addition on Hom(V,W) by (f + g)(v) = f(v) + g(v) and scalar multiplication by (af)(v) = af(v.
If V is a vector space over a field K, define V* = Hom(V,K). This is called the dual space of V. If <v1,.....,vn> is a basis of V, show that for each I there is an vi* that is an element of V* satisfying vi*(vj) = d_ij (Kronecker delta function).
*Hopefully I typed this question out clear enough, I'm hoping that a real maverick of linear algebra happens upon this thread and understands the question*
The Attempt at a Solution
I've put a lot of thought into this problem and wanted to discuss my thoughts with someone who knows what's going on. So given vi* is an element of V*, we need find what mapping vi* must be such that vi*(vj) = d_ij (Kronecker delta function).
what if vi*(vj) = ( vj / |vj| )e_i where e_i is a unit vector in the I direction. The (vj / |vj|) will equal one, since this is the equation for normalizing a vector, and then multiplying this by e_i will be zero if j does not equal I.
But then I realized that multiplying by e_i will be zero only if I apply grand Schmitt's orthonormalization process first, right?
Anyone, I'm knew to the subject and hope somebody can somewhat understand my thoughts and provide some insight! Thanks PF