Lectures on Linear Algebra

A Primer on Linear Systems

Space of the column vectors over a field.
Geometric interpretation of consistent and inconsistent systems of equations.
Transformation of a system of equations to the echelon form by elementary operations.

Systems of Linear Equations: Theory and Practice

Rank of a matrix and the Kronecker-Capelli consistency condition.
Practical implementation of the general theorems on systems of linear equations.
An instructive example with comprehensive discussion.
Application to mechanics: Equilibrium of a set of masses on springs.
Solving systems of linear equations in Sage.

Eigenvalues and Eigenvectors

General solution of the eigenproblem in finite-dimensional vector spaces.
Application to the theory of systems of ordinary 1st order differential equations.
Similarity and diagonalization of matrices.

Unitary Spaces

Inner (scalar) product in complex and real spaces.
Definition and examples of unitary (complex) and Euclidean (real) spaces.
Schwarz inequality and its specific implementations.
Orthogonality of vectors. Orthogonal complement of a subspace.
Orthogonal and orthonormal basis of a unitary space.
Gram-Schmidt method for orthonormalizing a set of vectors and the QR decomposition.
Hermitian and unitary matrices vs Hermitian and unitary operators.
Properties of normal matrices and operators.

Problems in Linear Algebra

Problems from this Chapter (or similar ones) may occur on the Exam.

Indices and tables