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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/05%3A_Linear_Transformations/5.04%3A_Special_Linear_Transformations_in_Two_and_Three_Dimensions/5.4E%3A_Exercises_for_Section_5.4This page features exercises on deriving transformation matrices for linear operations in R2 and R3, focusing on rotations, reflections, projections, shears, and dilations....This page features exercises on deriving transformation matrices for linear operations in R2 and R3, focusing on rotations, reflections, projections, shears, and dilations. It includes specific angles for rotations and combinations of transformations, with answers demonstrating the relationships between trigonometric functions and geometric effects.
- 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_Matrix/6.1E%3A_Exercises_for_Section_6.1This page presents a collection of exercises on eigenvalues and eigenvectors, focusing on their properties under various matrix operations, including powers and scalar multiplication. It explores prac...This page presents a collection of exercises on eigenvalues and eigenvectors, focusing on their properties under various matrix operations, including powers and scalar multiplication. It explores practical calculations for specific 3×3 matrices and theoretical discussions on transformations such as rotations, reflections, and projections. Key findings include conditions for eigenvalue existence and behaviors of complex eigenvalues.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.02%3A_The_Matrix_of_a_Linear_Transformation_I/5.2E%3A_Exercises_for_Section_5.2This page discusses linear transformations T in Rn with examples illustrating how they alter vector components and the associated matrices A. It covers transformations in \(\math...This page discusses linear transformations T in Rn with examples illustrating how they alter vector components and the associated matrices A. It covers transformations in R2, detailing specific cases like rotation and scaling, and describes the conditions under which a transformation matrix can be derived from vectors when an inverse exists.
- https://math.libretexts.org/Courses/Mount_Royal_University/Mathematical_Reasoning/4%3A_Basic_Concepts_of_Euclidean_Geometry/4.4%3A_TransformationsA translation is a transformation that moves a figure (without altering dimensions) to a new position.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/03%3A_Linear_Transformations_and_Matrix_Algebra/3.01%3A_Matrix_TransformationsThis page provides an overview of matrix transformations in linear algebra, emphasizing their geometric interpretation in R2 and their applications in robotics and computer graphics. It ...This page provides an overview of matrix transformations in linear algebra, emphasizing their geometric interpretation in R2 and their applications in robotics and computer graphics. It discusses key concepts such as domain, codomain, range, and the identity transformation while illustrating various transformations like rotation, shear, and projection.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Interactive_Linear_Algebra_(Margalit_and_Rabinoff)/05%3A_Eigenvalues_and_Eigenvectors/5.04%3A_Complex_EigenvaluesThis page discusses the study of matrices with complex eigenvalues, detailing the methods for calculating eigenvalues and eigenvectors, particularly for 2×2 and 3×3 matrices. It ...This page discusses the study of matrices with complex eigenvalues, detailing the methods for calculating eigenvalues and eigenvectors, particularly for 2×2 and 3×3 matrices. It highlights the significance of the characteristic polynomial, the rotation-scaling theorem, and the block diagonalization theorem, emphasizing geometric interpretations of transformations.
- https://math.libretexts.org/Courses/Las_Positas_College/Math_for_Liberal_Arts/09%3A_Geometry/9.04%3A_Geometric_Symmetry_and_TessellationsA rigid motion is the action of taking a geometric object in the plane and moving it is some fashion to another position in the plane without changing its shape or size. So if you make six copies of a...A rigid motion is the action of taking a geometric object in the plane and moving it is some fashion to another position in the plane without changing its shape or size. So if you make six copies of a single triangle and put them together at a point so that each angle appears twice, there will be a total of 360° around the point, meaning the triangles fit together perfectly with no gaps and no overlaps.
