These are lecture notes by Tao for a course in UCLA 2002, on the overview Tao writes:«This course is an introduction to Linear algebra. [...] Linear algebra is the study of the algebraic properties of linear transformations (and matrices). Algebra is concerned with how to manipulate symbolic combinations of objects, and how to equate one such combination with another; e.g. how to simplify an expression such as (x − 3)(x + 5). In linear algebra we shall manipulate not just scalars, but also vectors, vector spaces, matrices, and linear transformations.«These manipulations will include familiar operations such as addition, multiplication, and reciprocal (multiplicative inverse), but also new operations such as span, dimension, transpose, determinant, trace, eigenvalue, eigenvector, and characteristic polynomial. [Algebra is distinct from other branches of mathematics such as combinatorics (which is more concerned with counting objects than equating them) or analysis (which is more concerned with estimating and approximating objects, and obtaining qualitative rather than quantitative properties).]»