Do you want to learn how to work with abstract spaces and transformations that preserve their structure? Do you want to understand the concepts of vector subspaces, linear combinations, linear dependence, basis, dimension, coordinates, and properties of vector spaces? Do you want to master the skills of defining and manipulating linear maps, their kernels, images, matrix representations, and transition matrices? If you answered yes to any of these questions, then this course is for you! Linear Algebra: Linear Vector Spaces and Linear Maps is a comprehensive and engaging course that covers the fundamentals of vector spaces and linear maps and their applications in mathematics and science. You will learn how to: - Define and classify vector spaces and their subspaces over a given scalar field - Perform operations on vectors using linear combinations and scalar multiplication - Determine whether a set of vectors is linearly dependent or independent and find a basis and dimension for a vector space or subspace - Find the coordinates of a vector with respect to a given basis and change the basis using transition matrices - Define and classify linear maps between vector spaces and find their domains, codomains, ranges, and null spaces - Find the kernel and image of a linear map and use them to determine whether a linear map is one-to-one or onto - Represent a linear map using a matrix and perform matrix operations such as addition, multiplication, and inversion - Use different methods and tools to solve systems of linear equations, such as Gaussian elimination, row reduction, and inverse matrices This course is suitable for anyone who wants to learn or review the basics of vector spaces and linear maps and their applications. It is especially useful for students and professionals in algebra, geometry, analysis, differential equations, optimization, cryptography, computer graphics, data science, and other related fields. By the end of this course, you will have a solid foundation of the theory and practice of vector spaces and linear maps and their operations. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve vector spaces and linear maps. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

A comprehensive study of linear transformations (or maps) - definition, kernel, range, matrix representation and related matters.

Do you want to learn how to work with matrices and their properties, operations, and applications? Do you want to understand the concepts of determinants, eigenvalues, eigenvectors, diagonalization, quadratic and canonical forms? Do you want to master the skills of solving systems of linear equations, finding inverses, and computing matrix functions using different methods and tools? If you answered yes to any of these questions, then this course is for you! This course covers the fundamentals of matrix theory and its applications in mathematics and science. You will learn how to: - Define and classify matrices and their special types, such as symmetric, orthogonal, diagonal, and identity matrices - Perform matrix addition, subtraction, multiplication, and scalar multiplication using the algebraic properties of matrices - Find the transpose, conjugate, and adjoint of a matrix and use them to simplify matrix operations and expressions - Perform elementary row and column transformations on matrices and use them to find the row echelon form, reduced row echelon form, rank, and nullity of a matrix - Find the minors, cofactors, and determinants of matrices and use them to calculate the area, volume, and orientation of geometrical figures - Find the inverse of a matrix using the adjoint method or the row operations method and use it to solve systems of linear equations - Find the eigenvalues and eigenvectors of a matrix using the characteristic polynomial and the Cayley-Hamilton theorem and use them to analyze the behavior and stability of dynamical systems - Diagonalize a matrix using the eigenvalues and eigenvectors and use it to compute matrix functions, such as polynomials, exponentials, sines, and cosines of matrices - Find the quadratic and canonical forms of an equation using the coefficient matrix and the transformation matrix and use them to classify and graph conics and quadrics - Use computer software, such as MS-Excel, MATLAB, and Python, to perform matrix operations and computations efficiently and accurately This course is suitable for anyone who wants to learn or review the basics of matrix theory and its applications. It is especially useful for students and professionals in algebra, calculus, differential equations, linear programming, optimization, cryptography, computer graphics, data science, machine learning, and other related fields. By the end of this course, you will have a firm grasp of the theory and applications of matrices and determinants. You will also be able to apply the knowledge and skills you gain to solve real-world problems and challenges that involve matrices. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.