  Let $n>m$, and let $A$ be an $(m\times n)$-matrix of full rank. Then
obviously the estimate $\|Ax\|\leq\|A\|\|x\|$ holds for the euclidean norm of
$x$ and $Ax$ and the spectral norm as the assigned matrix norm. We study the
sets of all $x$ for which, for fixed $\delta<1$, conversely
$\|Ax\|\geq\delta\,\|A\|\|x\|$ holds. It turns out that these sets fill, in the
high-dimensional case, almost the complete space once $\delta$ falls below a
bound that depends on the extremal singular values of $A$ and on the ratio of
the dimensions. This effect has much to do with the random projection theorem,
which plays an important role in the data sciences. As a byproduct, we
calculate the probabilities this theorem deals with exactly.
