We describe the key technique of row reduction which both underlies efficient computer programs for solving systems of linear equations, and also provides a theoretical understanding. The appropriate theoretical framework is that of linear transformations acting on vector spaces. These are represented in practice by matrices, and we therefore study the algebra of matrices. By choosing the correct co-ordinate system we can often simplify the algebra. To do this we introduce the ideas of eigenvectors and eigenvalues and show how they can be used to diagonalize matrices.
Linear algebra provides a basis for wide areas of mathematics.
Linear equations, use of matrix notation, systematic row reduction, case of unique solutions, infinitely many solutions, and no solutions, geometrical interpretation of these cases. Rn as a vector space, linear independence, spanning, basis and dimension. Linear transformations, matrix representation of linear transformations, matrix multiplication, inverses of matrices, rank and nullity, the Dimension Theorem. Determinants, eigenvalues and eigenvectors. Postgraduate Module Catalogue Information set up and maintained by Geoff Lidster .