Linear Algebra
Linear algebra is the study of linear maps and the spaces they act on. Vectors are the objects; a matrix is the map — and almost every question in the subject is really a question about what that map does to space.
Read a matrix as a verb, not a table. Its columns record exactly where the basis vectors land, and that alone fixes the fate of every other vector — the grid deforms with them.
The first hard question is when a system Ax = b has a solution at all. Gaussian elimination answers it mechanically: clear each column below its pivot and the matrix steps into row-echelon form, where the solutions can simply be read off.
Beneath the mechanics sit the vector spaces that organize the answers. Independence, span, and basis measure how many directions a set of vectors really reaches, and dimension counts them.
Some directions survive a transformation untouched in orientation. An eigenvector is carried straight back onto its own line, merely stretched by an eigenvalue — the axes along which the map acts most simply.
Orthogonality turns geometry into computation. Projecting a vector onto a subspace drops a perpendicular to the closest point in it — the idea behind least squares, orthonormal bases, and the singular value decomposition.
Determinants measure how a map scales volume and whether it is invertible; an eigenbasis diagonalizes it; and the SVD factors any matrix into a rotation, a set of stretches, and another rotation.
The payoff is a single language for systems of equations, geometry, and data — and a set of factorizations that make each of them computable, even in floating point.