Fast Multiplication

Three applications of linear algebra [PDF]. Guest lecture of November 29, 2018 in Math 121. Mathematica notebook [.nb].
  x = 11234;
  y = 52342;
  X = Reverse[IntegerDigits[x]];
  Y = Reverse[IntegerDigits[y]];
  n = Length[X] + Length[Y] + 1;
  X = PadRight[X, n];
  Y = PadRight[Y, n];
  U = InverseFourier[X];
  V = InverseFourier[Y];
  Q = Round[Re[Fourier[U*V]*Sqrt[n]]];
  Sum[Q[[k]] 10^(k - 1), {k, n}]
   x*y
Update October 20, 2019: A news story in Popular mechanics featuring David Harvey who had been a graduate student here at Harvard. Was also discussed in Slashdot.