I've lately taken an interest in foundations of my field. While there are many important areas that contribute to Applied Mathematics (differential equations, probability & statistics, numerical methods, combinatorics), it seems that Linear Algebra is an essential ingredient in most Applied Mathematics projects. If one had to take only one course after the general calculus sequence, it seems like advanced Linear Algebra would be the choice, hands-down.

As examples:

In statistical applications, you often rely on solutions falling (at least approximately) in the space of Normal distributions, which is a linear space defined over the reals.

The differential operator is a linear operator, so you can represent systems of differential equations using linear algebra.

Numerical methods typically linearize a problem and then iterate to converge to a solution.

Is this just an artifact of my experiences, or is Linear Algebra really the foundational framework/theory of applied mathematics?

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    $\begingroup$ To the posters wanting to close this: Note the definition of "soft question" from Math SE: " Soft Question: For questions that don't admit a definitive answer but still are relevant to this site. Please be specific about what you are after." I have met this requirement. $\endgroup$ – user76844 Oct 16 '14 at 14:04
  • $\begingroup$ You might be interested in some of Gilbert Strang's books, especially his classic Introduction to Applied Mathematics in which he indeed presents linear algebra as being foundational to applied mathematics. $\endgroup$ – J W Mar 25 '15 at 15:43

“Statistical applications” and “normal distributions” also rely on error functions and the value of the Gaussian integral. Countless concepts in both economics and engineering represent the derivatives $($differentials$)$ or the anti-derivatives $($integrals$)$ of many other concepts. Etc.

  • $\begingroup$ Thanks for your answer. I am confused about what I am to conclude from this. Are you saying that Linear Spaces are ubiquitous in applications? $\endgroup$ – user76844 Oct 16 '14 at 14:08
  • $\begingroup$ @Eupraxis1981: No, I am saying that one might equally argue that Calculus is the mother and father / brick and mortar / alpha and omega / ground and foundation of applied mathematics, etc. $\endgroup$ – Lucian Oct 16 '14 at 14:12
  • $\begingroup$ @user76844 Pick up just about any book on numerical methods and see for yourself how much their algorithms depend of things being linear or locally linear. $\endgroup$ – steven gregory Mar 15 '16 at 3:29

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