When air pressure falls, bad weather often follows. High and low tides relate to the setting and rising of the moon. Hair and eye color pass from parent to child. Patterns emerge from systems as small as the atom and as large as the universe. Mathematics allow us to express these patterns as equations, and thereby to predict future changes.
Linear algebra provides the methods necessary to analyze unwieldy systems.
Insight into information such as the stresses and strains at multiple locations in a building, harmonic frequencies of an aircraft design, and how to maximize and minimize many individual aspects of an entire system allows the development of larger and more complex designs. Of special interest to industry, even after 40 years of intense research, is the ability to optimize. Before the 1940s, optimization techniques on systems of equations were essentially heuristics, simply educated guesses. Not until the Simplex Method was first proposed in 1947 by Dantzig did optimization become a defined process resulting in determinate answers. Optimization problems became linear programs, systems of equations that must be satisfied yet are underdetermined and thus allowing an infinite number of solutions.