Linear Algebra is where most of modern mathematics begins. Most people meet it first as a set of recipes: row-reduce this, invert that. But the arithmetic is not the real subject. The real subject is structure: the few rules a collection of objects must obey before we can add them, scale them, and map one to another without losing track of what we are doing. Once those rules are clear, the number-crunching can go to a computer, and what is left for us is the part that generalizes. That is why the same ideas keep coming back, unchanged in spirit, far from the grid of numbers where they started.
Within the Compass, Section I is the core. Much of what you study elsewhere on the site borrows its vocabulary from this section. The cryptography of Section IV (Discrete Mathematics & Algorithms) is built on the algebra of groups and fields. The smooth, infinite spaces of Section II (Calculus to Optimization & Analysis) are built on vector spaces and the maps between them. How the uncertain quantities of Section III (Probability & Statistics) vary together is recorded in matrices and their spectra. And the learning methods of Section V (Machine Learning) lean on all of these at once. Linear algebra is less one subject among many than the shared language the later sections all turn out to be speaking.
From this core, two roads open, and both are now built. The first follows symmetry. It starts with the rigid motions of a shape, moves to motions that vary continuously, such as the rotations of space, and then asks how such a group can act on a vector space. For compact groups such as the rotations, every finite-dimensional action splits into a direct sum of irreducible pieces, the way a sound splits into pure tones, and the road ends where Fourier series turn out to be one example of a much more general decomposition. The second road stays discrete. It passes from polynomials to finite fields, where every operation is exact. Beside them sit lattices, regular grids of points in space whose geometric problems are believed to be hard and now underlie encryption designed to resist quantum computers. Along the way, the state space of a quantum computer appears as what it is: a vector space, where each gate is a length-preserving linear map and measurement is the one step that is not. These roads look unrelated at first. Part of what this section shows is that they are the same few structures, meeting you again under different names.
Algebra is also the first place where we learn to tell things apart with certainty. An invariant is a quantity that stays the same under the changes we agree to ignore. The rank of a matrix survives any change of basis. The eigenvalues of a linear map do not depend on the coordinates we choose. The character of a representation is the same for any two isomorphic copies. If two objects disagree on an invariant, they are different, and no amount of searching will make them the same. Symmetry gives a second kind of guarantee. Suppose a network must turn a rotated input into a correspondingly rotated output. Representation theory says exactly which linear maps can do that, and if every layer, nonlinearities included, respects the rotation, the whole network does too. The symmetry then holds by design, and algebra can check it.