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SOURCE: Mikaelian V. Linear Algebra. Theory and Algorithms 2025
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Textbook in PDF format Dear Student, Welcome to Linear Algebra Introduction ! This is a one-semester course covering vectors, fields, systems of linear equations, matrices and determinants, abstract spaces, linear transformations, eigenvectors, diagonalization, inner products and related topics. These lecture notes are a learning instrument based on more then 20 years of experience in lecturing linear algebra. Since linear algebra is one of the most applicable areas of mathematics, special attention is paid to algorithms and problem solving methods. Having students from different departments, such as computer science, mathematics, engineering, business, etc., I know that their priorities may be different. Thus, below I always try to explain theory in detail even if I know some of the proofs are just optional martial for the Linear Algebra Introduction course. I feel safer to include all proofs, even if some students are going to skip the proofs to proceed to algorithms and examples. A general outline of the text: Part 1 starts with examples of real, rational, complex and modular spaces to prepare the student for the general concepts of field F, and of space F n over F. Some elements of analytic geometry in R2 and R3 are used to practice with vectors and operations on them. This part can well be omitted by students familiar with vectors, fields and their basic properties from school. Then in Part 2 and Part 3 the systems of linear equations and matrices are discussed on general fields. Gaussian row-elimination, row-echelon form and the reduced row-echelon form, factorization of matrices into elementary matrices are the main study tools in these parts. The accumulated material on fields, vectors, matrices, polynomials, etc., allows to define abstract vector spaces in Part 4. Bases and dimension of spaces, coordinate systems, change of basis in spaces are introduced. Then a series of computational methods are offered in Part 5. Part 6 is an introduction to determinants and their basic applications. The determinants earlier used to be central technical tools to study almost all other concepts of linear algebra. But later introduction of more efficient methods shifted attention awayfrom determinants. Part 7 introduces linear transformations and related objects, such as kernel and range of transformations, operations with transformations, etc. Study of transformations is continued in Part 8 using eigenvalues and eigenvectors. Diagonalization, invariant subspace, the Jordan normal form are considered. Part 9 is dedicated to abstract inner product spaces over real and complex fields, Gram matrix, orthogonal complement and projection. The method of least squares and regression are given as applications. Part 10 covers the main basic types of linear transformations in inner product spaces: the adjoint and normal transformations, orthogonal, unitary, symmetric and Hermitian transformations, real and complex Spectral theorems, positive definiteness and positive semidefiniteness, isometry and principal square root. Part 11 uses the basic transformation types from the previous part to build two types of general transformation decompositions, such as the polar decomposition and the singular value decomposition (SVD). As applications of this theory the r-approximation, image compression, pseudoinverse, least squares, linear regression, polynomial regression, etc. are given. In each chapter the theory is followed by algorithms and problem solving methods to accomplish certain types of tasks. The theory and the algorithms are illustrated by many examples
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