Permute transposition and derivative + Norm question

Question

I have two questions:

1. In general, what are the conditions to permute transposition and derivative: $$[x^{\prime}(t)]^{T}=[x^{T}(t)]^{\prime}$$

2. Is it correct to writ: $\quad$ $\mathbb{R}^{n}=\mathbb{M}_{n}(\mathbb{R}).\mathbb{R}^{n}$

   so if: $\quad$ $x=Py$

   then $\quad$$\|x\|_{\mathbb{R}^{n}}=\left\|P y\right\|_{\mathbb{R}^{n}}<\|P\|_{\mathbb{M}_{n}}\|y\|_{\mathbb{R}^{n}}$

Thank you for your justifications and explanations

Answer 1

1. In finite dimensional row and column vectors, transposition is linear (and bounded, continuous) and limits interchange with this operation. In more general vector spaces, you would have to define what transposition or something similar is, and in rare situations that operation might not be bounded=continuous.

2. Yes. As long as the matrix norm is associated to the vector norm. There are many different vector space norms and associated and unassociated matrix norms, and not for every combination do you get that inequality (which is not strict in general for the standard situation of associated norms).