This course, which assumes and builds upon a basic knowledge of matrix theory from MTH102, is designed to give a good grounding in all linear aspects of mathematics. The emphasis in sections A and B will be on actual examples and only basic results are proved. A more abstract approach is offered in the course in Algebra 1 (MTH005). The main aim in section C (which could be studied before section B) is to expose the link between matrix theory and linear transformations. This material, together with that in Section D, has applications to almost all areas of pure and applied mathematics. It is applied (within this course) to diagonalization of matrices and solutions of differential equations.
|(A) VECTOR SPACES. Basic definitions and examples, subspaces, dimension and basis, the vector spaces and subspaces of $\BbbR^n$, $n\in$ $\BbbN$.||5|
|(B) INNER PRODUCT SPACES. Basic definitions with many examples, the notions of norm (length) and distance (emphasis will be on the Euclidean inner product), the Cauchy-Schwarz inequality, the angle between two vectors, orthogonal vectors --- the Gram-Schmidt orthogonalization process, orthonormal basis.||3|
|(C) LINEAR TRANSFORMATIONS. Basic definitions and results, including images and kernels, with examples; matrix representations of linear transformations from $\BbbR^n$ to $\BbbR^n$, geometric interpretations of linear transformations from $\BbbR^2$ to $\BbbR^2$.||4|
|(D) EIGENVALUES AND EIGENVECTORS. The eigenvalues and eigenvectors of a matrix, the characteristic polynomial of a matrix, linear independence and orthogonality of eigenvectors, algebraic and geometric multiplicity of eigenvalues, eigenspaces, eigenvalues of powers of matrices, formulas for finding inverses of matrices and powers of matrices.||3|
|(E) VARIOUS TYPES OF MATRICES. Symmetric, skew symmmetric, unitary, hermitian matrices etc., their eigenvalues and eigenvectors, some basic results and theorems about them.||3|
|(F) DIAGONALISATION OF MATRICES. Similar matrices and their eigenvalues and vectors, identification of matrices that are diagonalizable, the Cayley-Hamilton theorem and its applications, quadratic forms and canonical forms.||3|
|(G) DIFFERENTIAL EQUATIONS. Application of diagonalization of matrices to solutions of systems of differential equations; the emphasis is on method --- no theorems are proved.||3|