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.
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?