# Linear Algebra: Course Outline For M.Sc Mathematics (MAT 402)

Linear Algebra is a branch of Mathematics which includes lines, planes and subspaces but it deals with properties of all common vector spaces

## Course Objective

At the end of this course the students will be able to setup and seek the solutions of several equations in several. Unlike other courses of its level, linear algebra embodies a circle of theoretical ideas which necessitate careful definitions, and statements and proofs of theorems, as well as a body of computational techniques that can serve both the theory itself and its application. Further they will be able to compute eigenvectors. The last part of the course will enable the students to work with and linear operators and their adjoints.

## Course Outline

• Review of matrices and determinants, Linear spaces, Bases and dimensions, Subspaces, Direct sums of subspaces, Factor spaces,
• Linear operators, Matrix representation and sums and products of linear operators, The range and null space of linear operators, Invariant subspaces.
• Eigen values and eigenvectors,
• Transformation to new bases and consecutive transformations, Transformations of the matrix of a linear operator,
• Canonical form of the matrix of a nilponent operator,
• Polynomial algebra and canonical form of the matrix of an arbitrary operator,
• The real Jordan canonical form,
• Bilinear and quadratic forms and reduction of quadratic form to a canonical form, Adjoint linear operators,
• Isomorphisms of spaces, Hermitian forms and scalar product in complex spaces,
• System of differential equations in normal form,
• Homogeneous linear systems, Solution by diagonalisation,  Non-homogeneous linear systems