Derivative of matrix Bilinear map

Question

Let A be an invertible $n\times n$matrix. Define a function $ F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ by $F(x,y)=<Ax,y>$. $<x,y>$ denotes inner product.Let DF(x,y) be derivative of F at (x,y) . Then

1. If $x\ne 0$ then $DF (x,0) \ne 0$
2. If $y\ne 0$ then $DF(0,y) \ne 0$
3. If $(x,y) \ne (0,0)$, then $DF(x,y) \ne 0$
4. If $x=0$ or $y=0$, then $DF(x,y)=0$

This question is about derivative of a bilinear form. But I know that derivative at (x,y) evaluated at some point (h,k) will be F(x,k)+F(h,y). But just derivative at (x,y) will be a linear tranformation. How to tackle this.

Answer 1

The Frechet derivative at $(h,k)$ is $DF(h,k)=\langle Ax,k\rangle+\langle Ah,y\rangle.$

Then, $DF(x,0)(h,k)=\langle Ax,k\rangle$ so clearly $DF(x,y)$ is not the $0\ \textit{transformation}$ if $x\neq 0.$

The others are proved similarly, except $4.$ should be $DF(0,0)=0.$