Derivative of $f(X)=X^T$

Question

Suppose that $X$ is a vector in $R^n$ and $f$ is a function that receives $X$ and gives $X^T$ which is the transpose. Is $f$ differentiable? If yes, what is its derivative?

Answer 1

Hint:

1. Prove that $f$ is a linear operator from $U=\mathbb{R}^{n\times 1}$ to $V=\mathbb{R}^{1\times n}$.
2. Prove that $f$ is a continuous linear operator from $U$ to $V$.
3. What is the Fréchet derivative of a continuous linear operator?