All Questions

Assume $V$ is a finite-dimensional vector space over $\mathbb{R}$, and $T: V \to V$ is a (linear) isomorphism. When is it possible to construct a norm on $V$ making $T$ an isometry? (Hopefully,...

Given two vector sets, $\vec x_i$ and $\vec y_i$ (for $i$=1,2,...N, but the dimensionality of each vector can be more than N), let their sum set be $\vec z_i = \vec x_i + \vec y_i$. It's easy to ...

Suppose I have a family of $n$ linearly-independent elements $v_i$ of the Hilbert space $\mathbb{C}^m$, which are not necessarily orthogonal. Can I always find a partial isometry $f: \mathbb{C} ^m \to ...