Linear algebra
Many equations share the same unknowns, and many quantities change together. Is there one language for solving them all at once and for seeing what a matrix does to space?
Where you are 11 inside
What this area is about
Linear algebra studies vectors, matrices and the maps between them that respect adding and scaling. It solves large linear systems by routine, and reads a matrix as a motion of space: what it reaches, what it crushes, and which directions it only stretches.
How these fit together
Tools. Matrices give the arithmetic and the determinant; row reduction solves any linear system; vectors in n dimensions bring span, independence and dimension.
The geometric reading. Matrices as transformations covers image, kernel and invertibility. Orthogonality and least squares finds closest points and best approximate solutions. Eigenvalues and eigenvectors finds the directions a square matrix only stretches.
Proof-based courses, not yet written. Abstract vector spaces: vector spaces from axioms. Linear maps: maps between vector spaces and their matrices. Eigenvalues, proved: invariant subspaces and diagonalizability. Inner product spaces: length and angle in any vector space. Structure of operators: generalized eigenvectors and Jordan form.
A path through it
Matrices, row reduction and matrices as transformations lean on one another, so they are taken together, matrices first. Vectors in \mathbb{R}^n follow matrix addition and come before transformations. Orthogonality and eigenvalues come after; eigenvalues needs transformations.
The proof-based courses follow in order: abstract vector spaces, linear maps, eigenvalues proved, then inner product spaces and structure of operators.