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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.07%3A_Basis_and_DimensionThis page discusses the concept of a basis for subspaces in linear algebra, emphasizing the requirements of linear independence and spanning. It covers the basis theorem, providing examples of finding...This page discusses the concept of a basis for subspaces in linear algebra, emphasizing the requirements of linear independence and spanning. It covers the basis theorem, providing examples of finding bases in various dimensions, including specific cases like planes defined by equations. The text explains properties of subspaces such as the column space and null space of matrices, illustrating methods for finding bases and verifying their dimensions.
- 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/07%3A_Vector_Spaces/7.07%3A_Sums_and_IntersectionsIn this section we discuss sum and intersection of two subspaces.
- 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/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/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/02%3A_Systems_of_Linear_Equations-_Geometry/2.08%3A_The_Rank_TheoremThis page explains the rank theorem, which connects a matrix's column space with its null space, asserting that the sum of rank (dimension of the column space) and nullity (dimension of the null space...This page explains the rank theorem, which connects a matrix's column space with its null space, asserting that the sum of rank (dimension of the column space) and nullity (dimension of the null space) equals the number of columns. It includes examples demonstrating how different ranks and nullities influence solution options in linear equations, emphasizing the theorem's importance in understanding the relationship between solution freedom and system properties without direct calculations.
- 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/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.06%3A_Directional_Derivatives_and_the_GradientA function z=f(x,y) has two partial derivatives: ∂z/∂x and ∂z/∂y. These derivatives correspond to each of the independent variables and can be interpreted as instantaneous rates of change ...A function z=f(x,y) has two partial derivatives: ∂z/∂x and ∂z/∂y. These derivatives correspond to each of the independent variables and can be interpreted as instantaneous rates of change (that is, as slopes of a tangent line). Similarly, ∂z/∂y represents the slope of the tangent line parallel to the y-axis. Now we consider the possibility of a tangent line parallel to neither axis.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/02%3A_Matrices/2.03%3A_Elementary_MatricesThis page covers the concept of elementary matrices, which are derived from the identity matrix using row operations. It details how these matrices are key in finding the inverse of matrices and expre...This page covers the concept of elementary matrices, which are derived from the identity matrix using row operations. It details how these matrices are key in finding the inverse of matrices and expresses a matrix as a product of elementary matrices. Properties of invertible matrices are discussed, including the conditions that an n×n matrix must meet to be invertible, emphasizing the significance of row operations.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/02%3A_Logic/2.04%3A_Mathematical_ProofThis page discusses the concept of proof in mathematics, emphasizing its role in understanding and communication. It covers two theorems about sets: one states that the intersection of two sets is a s...This page discusses the concept of proof in mathematics, emphasizing its role in understanding and communication. It covers two theorems about sets: one states that the intersection of two sets is a subset of either, and the other indicates that the set difference is part of the complement of the second set. Both theorems include detailed proofs, along with practice checkpoints for further investigation.
