  We present a simple solution to a question posed by Candes, Romberg and Tao
on the uniform uncertainty principle for Bernoulli random matrices. More
precisely, we show that a rectangular k*n random subgaussian matrix (with k <
n) has the property that by arbitrarily extracting any m (with m < k) columns,
the resulting submatrices are arbitrarily close to (multiples of) isometries of
a Euclidean space. We obtain the optimal estimate for m as a function of k,n
and the degree of "closeness" to an isometry. We also give a short and
self-contained solution of the reconstruction problem for sparse vectors.
