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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Book%3A_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/12%3A_Supplementary_notes_on_matrices_and_linear_systems/12.05%3A_Special_operations_on_matricesGiven positive integers m,n∈Z+ and any matrix A=(aij)∈Fm×n, we define the transpose AT=((aT)ij)∈Fn×m and the con...Given positive integers m,n∈Z+ and any matrix A=(aij)∈Fm×n, we define the transpose AT=((aT)ij)∈Fn×m and the conjugate transpose A∗=((a∗)ij)∈Fn×m by is orthogonal if A∈GL(n,R) and A−1=AT. Moreover, we define the (real) orthogonal group to be the set O(n)={A∈GL(n,R) | A−1=AT}.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/07%3A_Matrices/7.03%3A_Properties_of_MatricesThe objects of study in linear algebra are linear operators. We have seen that linear operators can be represented as matrices through choices of ordered bases, and that matrices provide a means of ef...The objects of study in linear algebra are linear operators. We have seen that linear operators can be represented as matrices through choices of ordered bases, and that matrices provide a means of efficient computation.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/10%3A_Linear_Systems_of_Differential_Equations/10.03%3A_Basic_Theory_of_Homogeneous_Linear_SystemsIn this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with...In this section we consider homogeneous linear systems y′=A(t)y, where A=A(t) is a continuous n×n matrix function on an interval (a,b). The theory of linear homogeneous systems has much in common with the theory of linear homogeneous scalar equations.
- https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_(Kravets)/01%3A_Vectors_in_Space/1.07%3A_Cylindrical_and_Quadric_SurfacesWe have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes...We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/12%3A_Matrices/12.02%3A_Properties_of_MatricesThe objects of study in linear algebra are linear operators. We have seen that linear operators can be represented as matrices through choices of ordered bases, and that matrices provide a means of ef...The objects of study in linear algebra are linear operators. We have seen that linear operators can be represented as matrices through choices of ordered bases, and that matrices provide a means of efficient computation.
- https://math.libretexts.org/Under_Construction/Purgatory/MAT-004A_-_Multivariable_Calculus_(Reed)/01%3A_Vectors_in_Space/1.07%3A_Cylindrical_and_Quadric_SurfacesWe have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes...We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/01%3A_Vectors_in_Space/1.06%3A_Quadric_SurfacesWe have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes...We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.
- https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_v2_(Reed)/12%3A_Vectors_in_Space/12.06%3A_Quadric_SurfacesWe have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes...We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q3/04%3A_Vectors_in_Space/4.07%3A_Quadric_SurfacesWe have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes...We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/12%3A_Vectors_in_Space/12.07%3A_Quadric_SurfacesWe have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes...We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/03%3A_Operations_on_Matrices/3.02%3A_The_Matrix_TraceIn this section we learn about a new operation called the trace. It is a different type of operation than the transpose. Given a matrix A , we can “find the trace of A ,” which is not a matrix but r...In this section we learn about a new operation called the trace. It is a different type of operation than the transpose. Given a matrix A , we can “find the trace of A ,” which is not a matrix but rather a number. We formally define it here.
