# 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.

## Algebra of Matrices¶

Matrix as a rectangular layout of elements from a field.
The $$\,m\times n\,$$ matrices over a field $$K$$ form the vector space over $$K$$.
Row and column rules defining product of two matrices.
Practical vector and matrix operations in Sage.

## Operations upon Matrices¶

Row-echelon and reduced row-echelon form of a matrix.
Elimination method applied to the augmented matrix of a system of linear equations.
Transpose of a matrix and its properties. Symmetric and skew-symmetric matrices.
Inverse of a matrix. Elementary matrices. Permutation matrices.
A practical algorithm for the matrix inversion by Gauss-Jordan elimination.

## Matrix Decompositions¶

LU decomposition of a matrix into the product of lower- and upper-triangular factors.

## 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.

## Linear Transformations¶

Properties of linear transformations.
Isomorphic vector spaces.
Matrices of linear transformations.
Change of basis and related formulae.

## 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.