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- 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/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)/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/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/04%3A_R/4.07%3A_Row_Column_and_Null_Spaces/4.7.E%3A_Exercise_for_Section_4.7This page presents exercises on matrices, emphasizing the calculation of bases for row, column, and null spaces, alongside ranks and nullities. It validates the Rank-Nullity Theorem and explores kerne...This page presents exercises on matrices, emphasizing the calculation of bases for row, column, and null spaces, alongside ranks and nullities. It validates the Rank-Nullity Theorem and explores kernel spaces as subspaces of Rn. Key topics include linearly independent rows, pivot columns, and methods for solving linear algebra problems.
- 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/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_AlgebraThis page explores the link between linear transformations and matrices, covering topics like matrix transformations, one-to-one and onto transformations, and their identification. It discusses matrix...This page explores the link between linear transformations and matrices, covering topics like matrix transformations, one-to-one and onto transformations, and their identification. It discusses matrix multiplication, including composition and scalar multiplication, and introduces matrix inverses for equation solving. Overall, it highlights the utility of linear algebra in analyzing transformations via matrices.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_SpacesThis page defines vector spaces and describes their properties, including operations, spanning sets, linear independence, and subspaces. It covers bases, subspace operations, linear transformations, a...This page defines vector spaces and describes their properties, including operations, spanning sets, linear independence, and subspaces. It covers bases, subspace operations, linear transformations, and the concepts of image and kernel. The text also discusses the matrix representation of linear transformations and introduces inner product spaces, which apply geometric concepts of length and orthogonality to general vector spaces.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.04%3A_Special_Linear_Transformations_in_Two_and_Three_DimensionsIn this section, we will examine some special examples of linear transformations in R2 including rotations, reflections., and projections.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.03%3A_Properties_of_Linear_TransformationsLet T:Rn↦Rm be a linear transformation. Then there are some important properties of T which will be examined in this section.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_TransformationsThis page covers linear transformations, including their properties and applications. It explains matrix multiplication in relation to transformations, details special types like rotations and project...This page covers linear transformations, including their properties and applications. It explains matrix multiplication in relation to transformations, details special types like rotations and projections, and characterizes transformations as one-to-one and onto. Key concepts such as isomorphisms, kernel, and image are introduced, along with methods for representing transformations across different bases and solving linear systems. The text includes exercises for practice in each section.
