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- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/06%3A_Functions/6.02%3A_More_about_FunctionsWe have also seen various ways to represent functions. We have also seen that sometimes it is more convenient to give a verbal description of the rule for a function. In cases where the domain and cod...We have also seen various ways to represent functions. We have also seen that sometimes it is more convenient to give a verbal description of the rule for a function. In cases where the domain and codomain are small, finite sets, we used an arrow diagram to convey information about how inputs and outputs are associated without explicitly stating a rule. In this section, we will study some types of functions, some of which we may not have encountered in previous mathematics courses.
- 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/Bookshelves/Linear_Algebra/Fundamentals_of_Matrix_Algebra_(Hartman)/03%3A_Operations_on_Matrices/3.01%3A_The_Matrix_TransposeThe transpose of a matrix is an operator that flips a matrix over its diagonal. Transposing a matrix essentially switches the row and column indices of the matrix.
- https://math.libretexts.org/Courses/Irvine_Valley_College/Math_26%3A_Introduction_to_Linear_Algebra/02%3A_Linear_Transformations_and_Matrix_Algebra/2.05%3A_Determinants-_DefinitionLearn the definition of the determinant. Learn some ways to eyeball a matrix with zero determinant, and how to compute determinants of upper- and lower-triangular matrices. Learn the basic properties ...Learn the definition of the determinant. Learn some ways to eyeball a matrix with zero determinant, and how to compute determinants of upper- and lower-triangular matrices. Learn the basic properties of the determinant, and how to apply them. Recipe: compute the determinant using row and column operations. In this section, we define the determinant, and we present one way to compute it. Then we discuss some of the many wonderful properties the determinant enjoys.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.01%3A_Matrix_Operations/2.1E%3A_Exercises_for_Section_2.1This page presents various exercises on matrix operations, including addition, multiplication, conditions for operation definitions, and properties of matrices. It covers topics like commuting matrice...This page presents various exercises on matrix operations, including addition, multiplication, conditions for operation definitions, and properties of matrices. It covers topics like commuting matrices, zero products, and the distinction between symmetric and skew-symmetric matrices. Key exercises involve calculating specific matrix entries, demonstrating the uniqueness of zero matrices, and expressing matrices as sums of symmetric and skew-symmetric components.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/06%3A_Functions/6.02%3A_More_about_FunctionsWe have also seen various ways to represent functions. We have also seen that sometimes it is more convenient to give a verbal description of the rule for a function. In cases where the domain and cod...We have also seen various ways to represent functions. We have also seen that sometimes it is more convenient to give a verbal description of the rule for a function. In cases where the domain and codomain are small, finite sets, we used an arrow diagram to convey information about how inputs and outputs are associated without explicitly stating a rule. In this section, we will study some types of functions, some of which we may not have encountered in previous mathematics courses.