Linear Algebra

Linear algebra is not a bag of formulas but a single language for one question: what does a transformation do to space? While a matrix is only a table of numbers that question has no answer; read it as an action on vectors and algebraic properties gain geometric meaning. The course follows the order the ideas depend on one another: vectors first — linear combinations, span, bases, dimension; then matrices, the table of a transformation; then determinants, the number that says whether a transformation collapses a direction; then systems of linear equations, where the shape of the solution set settles how many solutions exist; then eigenvalues and eigenvectors, exposing the directions a transformation leaves unchanged; and finally quadratic forms and positive definiteness. By the end you can read a matrix as a transformation, solve and interpret a linear system, decide whether it diagonalises, and say what a quadratic form does.