math.NA Numerical analysis
Linear programming algorithms are used in compressed sensing, which is now being used in MRI and imaging to increase resolution and/or decrease the number of measurements required.
Numerical analysis is what makes calculators work. (And so much more!)
We use numerical linear algebra to approximate solutions to discretized versions of complicated PDEs.
At the heart of Google's Pagerank algorithm is a relatively simple numerical eigenvector computation called the Power Method. The study of large complicated networks (e.g. Facebook) is done using tools from graph theory which again comes back to using the tools of numerical linear algebra.