Let $X$ be a finite-dimensional inner product space, and $T$ a linear operator on $X$. Let $W$ be a subset of $X$ with the following property:

If $T$ preserves norms on $W$, then $T$ is orthogonal on $X$.

One might say that the set $W$ 'detects' orthogonality.

My question is: what is the smallest number of elements that $W$ may contain (as a function of dim$(X) = n$) ?

Quite simply, we can observe that $n + {n \choose 2}$ would form an upper bound for the minimal cardinality of $W$. However, I cannot think of a way to proof whether or not this upper bound is also a lower bound.

isometriclinear operators; I don't see how it's related to orthogonality. – Zen Harper Mar 2 '11 at 1:15