  The Forster transform is a method of regularizing a dataset by placing it in
{\em radial isotropic position} while maintaining some of its essential
properties. Forster transforms have played a key role in a diverse range of
settings spanning computer science and functional analysis. Prior work had
given {\em weakly} polynomial time algorithms for computing Forster transforms,
when they exist. Our main result is the first {\em strongly polynomial time}
algorithm to compute an approximate Forster transform of a given dataset or
certify that no such transformation exists. By leveraging our strongly
polynomial Forster algorithm, we obtain the first strongly polynomial time
algorithm for {\em distribution-free} PAC learning of halfspaces. This learning
result is surprising because {\em proper} PAC learning of halfspaces is {\em
equivalent} to linear programming. Our learning approach extends to give a
strongly polynomial halfspace learner in the presence of random classification
noise and, more generally, Massart noise.
