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Linear algebra plays a vital role in studying different types of real-world problems to study the behavior of the issues. A definition of linear algebra that might be a part of algebra is concerned with equations of the first degree. Thus, at the fundamental level, it involves the discussion of matrices and determinants and the solutions of systems of linear equations, which have a wide application in further discussion of this subject. Linear algebra is a subject that has found the broadest range of applications in all branches of mathematics, physical and social sciences, and engineering. It has a more significant application in information sciences and control theory. This book begins with a detailed discussion of matrix operation, its properties, and its applications in finding the solution of linear equations and determinants. This textbook entitled Linear Algebra with Its Applications is intended to study matrices, vector spaces, eigenvalue, eigenvectors, linear transformation methods, inner product spaces, diagonalizations, applications to conics and quadrics, canonical forms, and least squares problems. This book contains 11 chapters. Chapter 1 discusses the properties of matrices and matrix operations needed to study solutions of systems of linear equations and determinants. Chapter 2 discusses the system of linear equations and its solution using the Gaussian elimination method, the Gauss–Jordan elimination method, and LU decomposition methods, and the definitions of determinants with their properties and Crammer’s rule. In contrast, Chapter 3 starts with a discussion of vector spaces in n-dimensional vector spaces that include the properties of a vector space and subspaces, linear combinations and spanning a vector space, finitely generated vector spaces, linear dependence and independence, basis and dimensions, rank, sum and intersection of subspaces, direct sums of subspaces, and more than two subspaces. Chapter 4 discusses the properties of eigenvalues and eigenvectors and some properties of inner product spaces, including the Gram–Schmidt orthogonalization process and QR-factorization. Chapter 5 discusses linear transformation, which includes Range and Kernel of transformation, one-to-one and invertible transformations, ordination representation of vectors, change of basis, isomorphism, transformations in computer graphics, and fractal pictures of nature. Chapter 6 discusses inner product spaces that include the Cauchy–Schwarz inequality, orthogonal complements, orthogonal sets and bases, projection of one vector onto another vector, orthogonal matrix theorem, the Gram–Schmidt orthogonalization process, projection of a vector onto a subspace, distance of a point from a subspace, and QR-factorization. Chapter 7 discusses the matrix representation of linear transformations along with the importance of matrix representation, visualization of the matrix representation, and the relation between matrix representations. In contrast, Chapter 8 covers the diagonalizations that include minimal polynomials, the Cayley–Hamilton theorem, power of a matrix, diagonal matrix representation of a linear operator, diagonalization of matrices, diagonalization of symmetric matrices, and orthogonal diagonalization. Chapter 9 discusses the application to conics and quadrics that covers quadratic forms, conics, quadrics, definite quadratic form, bilinear form, matrix representation of bilinear forms, symmetric and skew-symmetric bilinear form, symmetric bilinear forms and quadratic forms, eigenvalues of congruent matrices, Sylvester’s law of inertia, skew-symmetric bilinear form, and the application to the reduction of quadrics. Chapter 10 discusses the canonical forms that include triangularizable matrices, block triangular matrices, block diagonalization, Hermitian matrices, unitary matrix, Schur’s theorem, spectral theorem, normal matrices, nilpotent operators, Jordan canonical form, rational canonical form, minimum polynomial and Jordan canonical form, Jordan normal form, properties of Jordan matrix, and minimum polynomial of Jordan normal form While the last chapter, Chapter 11, discusses the least square problems that cover approximation of functions, Fourier approximation, least square solutions, least square curves, eigenvalues by iteration and connectivity of networks, the power method for an n × n matrix, difficulties in the solution of the system of equations, the condition number of a matrix, and the coding theory. This book is based on syllabi of linear algebra prescribed for undergraduate and postgraduate mathematics students in different institutions and universities in India and abroad. This book will be helpful for competitive examinations as well.