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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.02%3A_Orthogonal_ComplementsThis page explores orthogonal complements in linear algebra, defining them as vectors orthogonal to a subspace W in Rn. It details properties, computation methods (such as using RREF...This page explores orthogonal complements in linear algebra, defining them as vectors orthogonal to a subspace W in Rn. It details properties, computation methods (such as using RREF), and visual representations in R2 and R3. Key concepts include the relationship between a subspace and its double orthogonal complement, the equality of row and column ranks of matrices, and the significance of dimensions in relation to null spaces.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.12%3A_Transformation_of_Nonlinear_Equations_into_Separable_EquationsThis section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
- https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/03%3A_Vector_Calculus/3.07%3A_Surface_IntegralThis page explains surface integrals for both scalar and vector fields over parametric surfaces, emphasizing the importance of parameterization and orientations. It describes calculating surface area,...This page explains surface integrals for both scalar and vector fields over parametric surfaces, emphasizing the importance of parameterization and orientations. It describes calculating surface area, mass flux, and heat flow using integrals, providing examples and exercises. The text covers the distinction between orientable and nonorientable surfaces and the role of normal vectors.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/04%3A__Number_Representation_and_Calculation/4.03%3A__Addition_and_Subtraction_in_Base_SystemsThe document focuses on arithmetic in different numeral systems, specifically bases 2 through 9 and 12. It explains how computers use base 2 (binary) for calculations and how conventional base 10 arit...The document focuses on arithmetic in different numeral systems, specifically bases 2 through 9 and 12. It explains how computers use base 2 (binary) for calculations and how conventional base 10 arithmetic changes with non-decimal bases. It details the construction of addition tables for these bases and illustrates addition and subtraction through examples. Key examples include calculations in bases 6, 7, and 12, and it also highlights common errors.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/06%3A_Orthogonality/6.03%3A_Orthogonal_ProjectionThis page explains the orthogonal decomposition of vectors concerning subspaces in Rn, detailing how to compute orthogonal projections using matrix representations. It includes methods f...This page explains the orthogonal decomposition of vectors concerning subspaces in Rn, detailing how to compute orthogonal projections using matrix representations. It includes methods for deriving projection matrices, with an emphasis on linear transformations and their properties. The text outlines the relationship between a subspace and its orthogonal complement, utilizing examples to illustrate projection calculations and reflections across subspaces.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.03%3A_Spanning_SetsIn this section we will examine the concept of spanning introduced earlier in terms of Rn . Here, we will discuss these concepts in terms of abstract vector spaces.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.01%3A_Eigenvalues_and_Eigenvectors_of_a_MatrixSpectral Theory refers to the study of eigenvalues and eigenvectors of a matrix. In this section we consider what eigenvalues and eigenvectors are and how to find them.
