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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Understanding_Linear_Algebra_(Austin)This book arises from my belief that linear algebra, as presented in a traditional undergraduate curriculum, has for too long lived in the shadow of calculus. Many mathematics programs currently requi...This book arises from my belief that linear algebra, as presented in a traditional undergraduate curriculum, has for too long lived in the shadow of calculus. Many mathematics programs currently require their students to complete at least three semesters of calculus, but only one semester of linear algebra, which often has two semesters of calculus as a prerequisite.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/01%3A_Systems_of_Linear_Equations-_AlgebraThis page discusses the algebraic study of linear equations, detailing methods for solving them, particularly through row reduction. It explains a systematic approach to solving equations and how to e...This page discusses the algebraic study of linear equations, detailing methods for solving them, particularly through row reduction. It explains a systematic approach to solving equations and how to express solutions in parametric form. The content is organized into sections that build foundational knowledge on linear equations, algorithms for solutions, and solution representation.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.01%3A_Eigenvalues_and_EigenvectorsThis page explains eigenvalues and eigenvectors in linear algebra, detailing their definitions, significance, and processes for finding them. It discusses how eigenvectors result from matrix transform...This page explains eigenvalues and eigenvectors in linear algebra, detailing their definitions, significance, and processes for finding them. It discusses how eigenvectors result from matrix transformations and the linear independence of distinct eigenvectors. The text covers specific examples, including eigenvalue analysis for specific matrices and the conditions for eigenvalues, including zero.
- 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/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.03%3A_DiagonalizationThis page covers diagonalizability of matrices, explaining that a matrix is diagonalizable if it can be expressed as A=CDC−1 with D diagonal. It discusses the Diagonalization Theorem, eig...This page covers diagonalizability of matrices, explaining that a matrix is diagonalizable if it can be expressed as A=CDC−1 with D diagonal. It discusses the Diagonalization Theorem, eigenspaces, eigenvalues, and the significance of linear independence among eigenvectors. Multiple diagonal forms can arise, while geometric and algebraic multiplicities influence diagonalizability.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.02%3A_One-to-one_and_Onto_TransformationsThis page discusses the concepts of one-to-one and onto transformations in linear algebra, focusing on matrix transformations. It defines one-to-one as each output having at most one input and outline...This page discusses the concepts of one-to-one and onto transformations in linear algebra, focusing on matrix transformations. It defines one-to-one as each output having at most one input and outlines examples and theorems related to this property. The text emphasizes that a transformation is onto if every output corresponds to some input.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/06%3A_Systems_of_ODEs/6.02%3A_Linear_systems_of_ODEs/6.2E%3A_Exercises_for_Section_6.2This page details exercises focused on systems of differential equations, including verifying solutions, checking the linear independence of vectors, and matrix representations. Key tasks involve find...This page details exercises focused on systems of differential equations, including verifying solutions, checking the linear independence of vectors, and matrix representations. Key tasks involve finding general solutions, validating specific solutions for given matrices, and deriving forms with mathematical justification. The exercises explore various systems and functions, necessitating in-depth understanding and presentation of results linked to independence and matrix notation.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/03%3A_Determinants/3.02%3A_Properties_of_Determinants/3.2E%3A_Exercises_for_Section_3.2This page includes exercises on matrix operations, specifically focusing on determinants. It explains how row and column operations affect determinants, discusses properties linked to nilpotent and or...This page includes exercises on matrix operations, specifically focusing on determinants. It explains how row and column operations affect determinants, discusses properties linked to nilpotent and orthogonal matrices, and provides proofs regarding matrix similarities that maintain determinant values.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.09%3A_The_General_Solution_of_a_Linear_SystemIn this section we see how to use linear transformations to solve linear systems of equations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.04%3A_Applications_of_Spectral_Theory/6.4E%3A_Exercises_for_Section_6.4This page features exercises on diagonalizing matrices and solving initial value problems for differential equations. It includes tasks such as computing matrix powers after diagonalization and solvin...This page features exercises on diagonalizing matrices and solving initial value problems for differential equations. It includes tasks such as computing matrix powers after diagonalization and solving first-order systems through matrix exponentiation. Each exercise outlines problem setups, provides hints, and sometimes offers detailed solution computations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.04%3A_Applications_of_Spectral_TheoryThis section considers some applications of matrix diagonalization.
