# Lectures on Linear Algebra¶

## Introduction¶

## 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.*

*Matrix addition and scalar multiplication.*

*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.*

## Determinants¶

## Matrix Decompositions¶

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

## Tensor (Kronecker) Product of Matrices¶

## 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.*

## Diagonalization of Matrices¶

## Proofs of Selected Theorems¶

## Problems in Linear Algebra¶

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