Question

prove E vector equals to -grad V
​

Answer 1

Explanation:

As an exercise in my textbook, I need to prove that if V is a finite dimensional vector space with dual space V∗ over R, then dim(V)=dim(V∗).

Let ω∈V∗ and let {e1,...,en} be a basis for V. Define ei∈V∗ by ei(ej)=δij. We show that {e1,...,en} spans V∗. ω(v)=ω(v1e1+...+vnen)=v1ω(e1)+...+vnω(en). If ω(e1)=λ1,...,ω(en)=λn, then ω(v)=v1λ1e1(e1)+...+vnλnen(en)=λ1e1(v)+...+λnen(v).

To show {e1,...,en} is linearly independent, suppose that 0=c1e1+...+cnen is the zero mapping to R. Consider the image of e1: 0(e1)=c1∗1+...+cn∗0=c1 Hence, c1=0. Repeating the procedure for ej, 2≤j≤n, we see that c1=c2=...=cn=0.