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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/06%3A_Spectral_Theory/6.02%3A_Eigenvalues_and_Eigenvectors_for_Special_MatricesIn this section we consider three kinds of matrices where we can simplify the process of finding eigenvalues and eigenvectors.
- 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/04%3A_R/4.06%3A_Subspaces_and_BasesThe goal of this section is to develop an understanding of a subspace of Rn.
- 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)/03%3A_Functions/3.05%3A_Proof_by_ContradictionThis page discusses direct proof and proof by contradiction, featuring a theorem that the composition of two functions is a function. The proof is divided into two parts: confirming that every domain ...This page discusses direct proof and proof by contradiction, featuring a theorem that the composition of two functions is a function. The proof is divided into two parts: confirming that every domain element maps to a codomain element, and using contradiction to establish uniqueness in mapping. It also introduces mapping composition and explains its operation with two mappings.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/02%3A_Applications_of_Integration/2.09%3A_Calculus_of_the_Hyperbolic_FunctionsThis page discusses differentiation and integration of hyperbolic functions and their inverses, emphasizing their calculus applications, particularly in modeling catenary curves. Key objectives includ...This page discusses differentiation and integration of hyperbolic functions and their inverses, emphasizing their calculus applications, particularly in modeling catenary curves. Key objectives include understanding derivatives, integrals, and their respective formulas for hyperbolic functions, as well as domain considerations for inverse functions. The text provides examples, exercises, and instructions for evaluating integrals using u-substitution.
- 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/Courses/Cosumnes_River_College/Math_372%3A_College_Algebra_for_Calculus/10%3A_Appendix_-_The_Language_of_Mathematics/10.02%3A_Axioms_Theorems_and_ProofsTo determine the assumption and conclusion of a theorem, it is recommended to rewrite the theorem in the form "If..., then...". When written in this form, the statement following the word "if" is the ...To determine the assumption and conclusion of a theorem, it is recommended to rewrite the theorem in the form "If..., then...". When written in this form, the statement following the word "if" is the assumption, and the statement following the word "then" is the conclusion. Thus, the assumption is that Camille is speeding down the street, and the conclusion is the consequence of that action, which is that she will be ticketed.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/08%3A_Appendices/8.02%3A_Well_Ordering_and_InductionThis page introduces summation notation and its applications, emphasizing well-ordered sets and mathematical induction. It explains how summation notation provides a concise representation of sums and...This page introduces summation notation and its applications, emphasizing well-ordered sets and mathematical induction. It explains how summation notation provides a concise representation of sums and describes the principle of well-ordering underlying induction. The section outlines the induction process, including base cases and steps, illustrated by examples that prove formulas and inequalities for all natural numbers.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.09%3A_Gram-Schmidt_ProcessThe Gram-Schmidt process is an algorithm to transform a set of vectors into an orthonormal set spanning the same subspace, that is generating the same collection of linear combinations.
