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- https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/02%3A_Matrices/2.01%3A_Matrix_ArithmeticYou 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.
- https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/08%3A_Systems_of_Equations_and_Matrices/8.02%3A_Systems_of_Linear_Equations-_Augmented_MatricesWe previously introduced Gaussian Elimination as a means of transforming a system of linear equations into triangular form with the ultimate goal of producing an equivalent system of linear equations ...We previously introduced Gaussian Elimination as a means of transforming a system of linear equations into triangular form with the ultimate goal of producing an equivalent system of linear equations which is easier to solve. If we study the process, we see that all of our moves are determined entirely by the coefficients of the variables involved, and not the variables themselves. In this section, we introduce a bookkeeping device to help us solve systems of linear equations.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/06%3A_Systems_of_ODEs/6.04%3A_Matrices_and_linear_systems/6.4E%3A_Exercises_for_Section_6.4This page features an exercise set focused on matrix operations, including solving systems of equations with matrix inverses, computing determinants, and exploring matrix invertibility conditions. It ...This page features an exercise set focused on matrix operations, including solving systems of equations with matrix inverses, computing determinants, and exploring matrix invertibility conditions. It includes tasks like determining invertibility for matrices with variables, verifying matrix relationships, and provides solutions for all exercises. Key examples focus on determinant calculation and identifying nonzero matrices that meet specific multiplication criteria.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.02%3A_The_Inverse_of_a_MatrixThis page explores matrix operations, focusing on the identity matrix and matrix inverses, including their existence, uniqueness, and the method for finding inverses through augmented matrices and row...This page explores matrix operations, focusing on the identity matrix and matrix inverses, including their existence, uniqueness, and the method for finding inverses through augmented matrices and row operations. It provides examples illustrating both the derivation of inverses and scenarios where matrices lack inverses.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.02%3A_Properties_of_DeterminantsThere are many important properties of determinants. Since many of these properties involve the row operations discussed in Chapter 1, we recall that definition now. We will now consider the effect of...There are many important properties of determinants. Since many of these properties involve the row operations discussed in Chapter 1, we recall that definition now. We will now consider the effect of row operations on the determinant of a matrix. In future sections, we will see that using the following properties can greatly assist in finding determinants. This section will use the theorems as motivation to provide various examples of the usefulness of the properties.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.05%3A_Markov_MatricesIn this section we look at a particular kind of matrix, called a Markov matrix, and consider some its applications.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.07%3A_Orthogonal_DiagonalizationIn this section we look at matrices that have an orthonormal set of eigenvectors.
- 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/04%3A_R/4.08%3A_Orthogonal_Vectors_and_Matrices/4.8.E%3A_Exercise_for_Section_4.8This page outlines exercises on determining orthogonality and orthonormality of vectors, classifying matrices (symmetric, skew symmetric, orthogonal), and the properties of orthogonal matrices, such a...This page outlines exercises on determining orthogonality and orthonormality of vectors, classifying matrices (symmetric, skew symmetric, orthogonal), and the properties of orthogonal matrices, such as preserving vector lengths.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_TheoryThis page discusses Eigenvalues and Eigenvectors in Spectral Theory, covering special matrices, diagonalization, applications, Markov matrices, dynamical systems, orthogonal diagonalization, singular ...This page discusses Eigenvalues and Eigenvectors in Spectral Theory, covering special matrices, diagonalization, applications, Markov matrices, dynamical systems, orthogonal diagonalization, singular value decomposition, special factorizations, and quadratic forms. It includes exercises for practice to enhance understanding of both theoretical and practical aspects of these concepts.
- 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.
