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- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.09%3A_The_General_Solution_of_a_Linear_SystemIn this section we see how to use linear transformations to solve linear systems of equations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.02%3A_Gaussian_EliminationThe work we did in the previous section will always find the solution to the system. In this section, we will explore a less cumbersome way to find the solutions, by representing a linear system as an...The work we did in the previous section will always find the solution to the system. In this section, we will explore a less cumbersome way to find the solutions, by representing a linear system as an augmented matrix and performing a systematic set of operations on it to reach a solution to the system.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.01%3A_Matrix_OperationsYou have now solved systems of equations by writing them in terms of an augmented matrix and then doing row operations on this augmented matrix. It turns out that matrices are important not only for s...You have now solved systems of equations by writing them in terms of an augmented matrix and then doing row operations on this augmented matrix. It turns out that matrices are important not only for systems of equations but also in many applications. In this section, we explore some matrix operations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.02%3A_Gaussian_Elimination/1.2E%3A_Exercises_for_Section_1.2This page contains exercises on augmented matrices and their impact on system solutions, focusing on consistency, uniqueness, and the influence of parameters. It details processes for row reducing mat...This page contains exercises on augmented matrices and their impact on system solutions, focusing on consistency, uniqueness, and the influence of parameters. It details processes for row reducing matrices to find row-echelon and reduced row-echelon forms, leading to potential solutions for linear equations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.03%3A_Application_of_the_Determinant_to_Inverses_Cramer's_Rule/3.3E%3A_Exercises_for_Section_3.3This page outlines exercises on matrix operations, including determining invertibility via determinants, calculating adjugates and inverses, and using Cramer’s Rule for solving equations. It reveals t...This page outlines exercises on matrix operations, including determining invertibility via determinants, calculating adjugates and inverses, and using Cramer’s Rule for solving equations. It reveals that matrix A is invertible, while matrix B is not, highlighting the role of determinants in unique solutions. Practical applications in electrical circuits are discussed along with challenges addressing numerical stability and conditions for unique solutions.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.01%3A_Systems_of_Linear_EquationsThis page discusses methods for solving systems of equations with two or three variables, covering unique, infinite, and no solutions. It emphasizes the significance of graphical representations, inte...This page discusses methods for solving systems of equations with two or three variables, covering unique, infinite, and no solutions. It emphasizes the significance of graphical representations, intersections of lines and planes, and the complexities introduced by additional variables. The page addresses consistent versus inconsistent systems, homogeneous systems, and the application of elementary operations that preserve solution sets.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/01%3A_Systems_of_Linear_Equations-_Algebra/1.01%3A_Systems_of_Linear_EquationsThis page introduces Rn as the set of ordered n-tuples of real numbers for labeling geometric points, focusing on linear equations' structure, consistency, and solutions. It discusse...This page introduces Rn as the set of ordered n-tuples of real numbers for labeling geometric points, focusing on linear equations' structure, consistency, and solutions. It discusses the geometric interpretation of solutions in n-dimensional space, illustrating how linear equations define lines or planes.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_MatricesThis page covers matrix operations crucial for solving systems of equations, including matrix inverses, the role of elementary matrices in row operations and theorems, and LU factorization into lower ...This page covers matrix operations crucial for solving systems of equations, including matrix inverses, the role of elementary matrices in row operations and theorems, and LU factorization into lower and upper triangular matrices. It includes exercises for practice.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.02%3A_Classification_of_Differential_Equations/1.2E%3A_Exercises_for_Section_1.2This page presents exercises centered on classifying and solving ordinary and partial differential equations. It emphasizes the detailed classification of equations based on order, linearity, and homo...This page presents exercises centered on classifying and solving ordinary and partial differential equations. It emphasizes the detailed classification of equations based on order, linearity, and homogeneity. Tasks include identifying equation types, providing examples of ODEs, discussing linearity conditions, and applying vector calculus concepts such as divergence and curl.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/06%3A_Systems_of_ODEs/6.03%3A_Eigenvalue_MethodIn this section we will learn how to solve linear homogeneous constant coefficient systems of ODEs by the eigenvalue method.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/01%3A_Systems_of_Linear_Equations-_Algebra/1.03%3A_Parametric_FormThis page explains parametric form and free variables in solving linear equations. It outlines how to express solution sets using free variables, demonstrating the infinite solutions available when at...This page explains parametric form and free variables in solving linear equations. It outlines how to express solution sets using free variables, demonstrating the infinite solutions available when at least one variable is free. The text also classifies systems of linear equations based on their augmented matrix forms, identifying three scenarios: inconsistent systems with no solutions, unique solutions, and systems with infinitely many solutions due to free variables.
