Some Linear Algebra....?

Question

Let V be a finite-dimesional space over a field K with a non-degenerate scalar product. Let A be a linear map from V to V. Show that the image of A-transpose is the orthogonal space to the kernel of A.

Answer 1

we use A* to denote the transpose of A .

(1) For all y in V , and x in Kernal of A
= = <0,y>=0


(2)Conversely suppose =0 for all y in V , then Ax=0
therefore x is in Kernal of A.

Combine the above two , we have our conclusion as your question mentioned .