Part of the challenge with linear algebra is that a lot of the basics are somewhat dry, since they serve mostly as a way to organize computation -- e.g., a syst

Part of the challenge with linear algebra is that a lot of the basics are somewhat dry, since they serve mostly as a way to organize computation -- e.g., a system of linear equations can be expressed as Ax <= b, where A is a matrix, b is a vector, and x is a vector of variables. Much of what makes linear algebra interesting and powerful comes from more advanced topics, especially eigenvalues. This power comes when we are not looking at a single matrix in isolation, but when we repeatedly apply a matrix. For instance, consider the equation x_t = A^t x_0. It turns out we can rewrite this an equation by diagonalizing A -- i.e., A = P^{-1} Sigma P, where Sigma is diagonal; most, but not all, matrices can be written in this form. We call s_i the "eigenvalues" of A. Then, the equation simplifies to x_t = P^{-1} Sigma^t P x_0, or equivalently (P x_t) = Sigma^t (P x_0). This equation is dramatically simpler, since Sigma is diagonal, so if Sigma = diag(s_1, ..., s_n), then Sigma^t = diag(s_1^t, ..., s_n^t). In other words, this transformation "disentangles" the different components of A into ones that act independently. Here, the transformation x -> P x is what is called a "change of basis". These repeated matrix applications are common in physics, where they represent how a dynamical system evolves over time. The main difference is that in physics, the system evolves continuously, but similar transformations can be applied to solve these problems.