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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.03%3A_Linear_TransformationsThis page covers linear transformations and their connections to matrix transformations, defining properties necessary for linearity and providing examples of both linear and non-linear transformation...This page covers linear transformations and their connections to matrix transformations, defining properties necessary for linearity and providing examples of both linear and non-linear transformations. It highlights the importance of the zero vector, standard coordinate vectors, and defines transformations like rotations, dilations, and the identity transformation.
- 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/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/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/05%3A_Graph_Theory/5.02%3A_Properties_of_GraphsThis page provides definitions and examples of graph properties like adjacency, vertex degrees, and types of graphs (regular, complete, bipartite). It covers subgraphs, graph complements, and duals, a...This page provides definitions and examples of graph properties like adjacency, vertex degrees, and types of graphs (regular, complete, bipartite). It covers subgraphs, graph complements, and duals, along with practice checkpoints for calculating degrees and understanding independent sets and maximum matchings. Each definition is illustrated with examples to aid in the comprehension of graph theory concepts.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/05%3A_Graph_Theory
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_DeterminantsThis page explains the importance of determinants in square matrices, covering their properties, applications, and geometric interpretations. It discusses their relation to matrix inverses and Cramer'...This page explains the importance of determinants in square matrices, covering their properties, applications, and geometric interpretations. It discusses their relation to matrix inverses and Cramer's Rule, emphasizes the effects of row operations on determinants, and includes examples for clarity. Furthermore, it highlights the geometric interpretation of determinants as volumes, enhancing comprehension of their defining properties and relevance in multivariable calculus.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Math_For_Liberal_Art_Students_2e_(Diaz)/03%3A_Sets/3.04%3A_CardinalityOften times we are interested in the number of items in a set or subset. This is called the cardinality of the set. The number of elements in a set is the cardinality of that set. The cardinality of ...Often times we are interested in the number of items in a set or subset. This is called the cardinality of the set. The number of elements in a set is the cardinality of that set. The cardinality of the set A is often notated as |A| or n (A).
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.04%3A_Matrix_MultiplicationThis page explores the interplay between compositions of transformations and matrix multiplication in linear algebra. It defines the composition of transformations, illustrates their properties, inclu...This page explores the interplay between compositions of transformations and matrix multiplication in linear algebra. It defines the composition of transformations, illustrates their properties, including non-commutativity and associativity, and connects these concepts to matrix operations. The Row-Column Rule for matrix multiplication is explained, alongside the implications of this non-commutative nature.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/04%3A_Determinants/4.01%3A_Determinants-_DefinitionThis page provides an extensive overview of determinants in linear algebra, detailing their definitions, properties, and computation methods, particularly through row reduction. It emphasizes the sign...This page provides an extensive overview of determinants in linear algebra, detailing their definitions, properties, and computation methods, particularly through row reduction. It emphasizes the significance of row operations, such as swaps and scaling, and introduces concepts like triangular matrices and multilinearity. Key properties include conditions for invertibility, the relationship between determinants of products, transposes, and the implications of zero determinants.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/04%3A_DeterminantsThis page discusses matrix equations, focusing on solving \(Ax=b\), determining eigenvalues and eigenvectors, and finding approximate solutions. The current chapter emphasizes determinants, covering t...This page discusses matrix equations, focusing on solving \(Ax=b\), determining eigenvalues and eigenvectors, and finding approximate solutions. The current chapter emphasizes determinants, covering their definition, properties, and computation methods. It includes cofactor expansions as a recursive calculation method and explores the geometric interpretation of determinants in relation to volumes in multivariable calculus.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/02%3A_Multiple_Integration/2.01%3A_Double_Integrals_over_Rectangular_RegionsIn this section we investigate double integrals and show how we can use them to find the volume of a solid over a rectangular region in the xyxy-plane. Many of the properties of double integrals are s...In this section we investigate double integrals and show how we can use them to find the volume of a solid over a rectangular region in the xyxy-plane. Many of the properties of double integrals are similar to those we have already discussed for single integrals.
