Is linear algebra laying the foundation for something important?

Linear Algebra is indeed one of the foundations of modern mathematics. There are a lot of things which use the language and tools developed in linear algebra:

• Multidimensional Calculus, i.e. Analysis for functions of many variables, i.e. vectors (for example, the first derivative becomes a matrix)

• Differential Geometry, which investigates structures which look locally like a vector space and functions on this.

• Functional Analysis, which is essentially linear algebra on infinite-dimensional vector spaces which is the foundation of quantum mechanics.

• Multivariate Statistics, which investigates vectors whose entries are random. For instance, to describe the relation between two components of such random vector, one can calculate the correlation matrix. Furthermore, one can apply a technique called singular value decomposition (which is close to calculating the eigenvalues of a matrix) to find which components are having a main influence on the data.

• Tagging on to the Multivariate Statistics and multidimensional calculus, there are a number of Machine Learning techniques which require you to find a (local) minimum of a nonlinear function (the likelihood function), for example for neural nets. Generally speaking, on can try to find the parameters which maximize the likelihood, e.g. by applying the gradient descent method, which uses vectors arithmetic. (Thanks, frogeyedpeas!)

• Control Theory and Dynamical Systems theory is mainly concerned with differential equations where matrices are factors in front of the functions. It helps tremendously to know the eigenvalues of the matrices involved to predict how the system will behave and also how to change the matrices in front to make sure the system behaves like you want it to - in Control Theory, this is related to the poles and zeros of the transfer function, but in essence it's all about placing eigenvalues at the right place. This is not only relevant for mechanical systems, but also for electric engineering. (Thanks, Look behind you!)

• Optimization in general and Linear Programming in particular is closely related to multidimensional calculus, namely about finding minima (or maxima) of functions, but you can use the structure of vector spaces to simplify your problems. (Thanks, Look behind you!)

On top of that, there are a lot of applications in engineering and physics which use tools of linear algebra and the fields listed above to solve real-world problems (often calculating eigenvalues and solving differential equations).

In essence, a lot of things in the mathematical toolbox in one variable can be lifted up to the multivariable case with the help of Linear Algebra.

Edit: This list is by no means complete, these were just the topics which came to my mind at first thought. Not mentioning a particular field doesn't mean that this field irrelevant, but just that I don't feel qualified to write a paragraph about it.

By now, we can roughly say that all we fully understand in Mathematics is linear.

So I guess Linear Algebra is a good topic to master.

Both in Mathematics and in Physics you tend to bring the problem to a Linear problem and then solve it with Linear Algebras techniques. This happens in Algebra with linear representations, in Functional Analysis with the study of Hilbert Spaces, in Differential Geometry with the tangent spaces, and so on almost in every field of Mathematics. Indeed I think there should be at least 3 courses of Linear Algebra (undergraduate, graduate, advanced), everytime with different insights on the subject.

Excellent answers above. I would just like to add:

• Computer graphics. As soon as you scale, rotate and/or translate an object on a computer screen, that is Linear Algebra in action. Computer graphics is an academic field that depend heavily on Linear Algebra.