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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Introduction_to_Matrix_Algebra_(Kaw)/01%3A_Chapters/1.05%3A_System_of_Equations\[\begin{bmatrix} a_{11} & a_{12} & \cdot & \cdot & a_{1n} \\ a_{21} & a_{22} & \cdot & \cdot & a_{2n} \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ a_{n1} & a_{...\[\begin{bmatrix} a_{11} & a_{12} & \cdot & \cdot & a_{1n} \\ a_{21} & a_{22} & \cdot & \cdot & a_{2n} \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot \\ a_{n1} & a_{n2} & \cdot & \cdot & a_{nn} \\ \end{bmatrix}\begin{bmatrix} a_{11}^{'} \\ a_{21}^{'} \\ \cdot \\ \cdot \\ a_{n1}^{'} \\ \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ \cdot \\ \cdot \\ 0 \\ \end{bmatrix} \nonumber \]
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.04%3A_Balancing_Chemical_ReactionsThe tools of linear algebra can also be used in the subject area of Chemistry, specifically for balancing chemical reactions.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.07%3A_Gauss-Seidel_Method/1.7E%3A_Exercises_for_Section_1.8This page provides exercises on the Gauss-Seidel method for solving systems of equations, detailing convergence criteria based on the coefficient matrix \(A\). It includes tasks for iterating using th...This page provides exercises on the Gauss-Seidel method for solving systems of equations, detailing convergence criteria based on the coefficient matrix \(A\). It includes tasks for iterating using the method with provided systems and initial guesses while calculating maximum absolute relative errors. The exercises conclude with approximate solutions after three iterations for various systems.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.05%3A_One-to-One_and_Onto_TransformationsThis section is devoted to studying two important characterizations of linear transformations, called One to One and Onto.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.05%3A_Applications_to_PhysicsThe tools of linear algebra can be used to study the application of resistor networks and dimensionless variables.
- 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_System/5.9E%3A_Exercises_for_Section_5.9This page discusses solutions to linear systems of equations in matrix form, providing detailed exercises along with their solution sets expressed as linear combinations of vectors. Each exercise illu...This page discusses solutions to linear systems of equations in matrix form, providing detailed exercises along with their solution sets expressed as linear combinations of vectors. Each exercise illustrates the relationship between homogeneous and non-homogeneous equations, presents the general solution format, and identifies the basis for the solution space.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.02%3A_The_Inverse_of_a_Matrix/2.2E%3A_Exercises_for_Section_2.2This page outlines exercises on matrix properties, particularly focusing on matrix inverses and related linear algebra concepts. It highlights critical examples, such as proving the identity matrix's ...This page outlines exercises on matrix properties, particularly focusing on matrix inverses and related linear algebra concepts. It highlights critical examples, such as proving the identity matrix's role, conditions for equality in matrix equations, and the existence of inverses. The exercises cover solving equations using matrix inverses, establishing unique inverses, and demonstrating properties of transposes and matrix products.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/01%3A_Systems_of_Equations/1.03%3A_Rank_and_Homogeneous_SystemsThere is a special type of system which requires additional study. This type of system is called a homogeneous system of equations. Our focus in this section is to consider what types of solutions are...There is a special type of system which requires additional study. This type of system is called a homogeneous system of equations. Our focus in this section is to consider what types of solutions are possible for a homogeneous system of equations. We also define two important terms: linear combination and rank.
