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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/07%3A_Linear_Transformations/7.03%3A_Isomorphisms_and_CompositionIf t1T(e1)+⋯+tnT(en)=0 with ti in R, then T(t1e1+⋯+tnen)=0, so \(t_{1}\m...If t1T(e1)+⋯+tnT(en)=0 with ti in R, then T(t1e1+⋯+tnen)=0, so t1e1+⋯+tnen=0 (because ker T={0}).
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/06%3A_Linear_Transformations/6.10%3A_Supplementary_Notes_-_More_on_Linear_Transformations/6.10.03%3A_Isomorphisms_and_CompositionIf t1T(e1)+⋯+tnT(en)=0 with ti in R, then T(t1e1+⋯+tnen)=0, so \(t_{1}\m...If t1T(e1)+⋯+tnT(en)=0 with ti in R, then T(t1e1+⋯+tnen)=0, so t1e1+⋯+tnen=0 (because ker T={0}).
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_(Morris)/03%3A_Graph_Theory/11%3A_Basics_of_Graph_Theory/11.04%3A_Graph_IsomorphismsThere is a problem with the way we have defined Kn. A graph is supposed to consist of two sets, V and E. Unless the elements of the sets are labeled, we cannot distinguish amongst them. Which of these...There is a problem with the way we have defined Kn. A graph is supposed to consist of two sets, V and E. Unless the elements of the sets are labeled, we cannot distinguish amongst them. Which of these graphs is K2 ? They can’t both be K2 since they aren’t the same graph – can they? The answer lies in the concept of isomorphisms.
- https://math.libretexts.org/Bookshelves/Applied_Mathematics/Seven_Sketches_in_Compositionality%3A_An_Invitation_to_Applied_Category_Theory_(Fong_and_Spivak)/03%3A_Databases-_Categories_functors_and_(co)limits/3.02%3A_CategoriesFor any graph G = (V, A, s, t), we can define a category Free(G), called the free category on G, whose objects are the vertices V and whose morphisms from c to d are the paths from c to d. The reason ...For any graph G = (V, A, s, t), we can define a category Free(G), called the free category on G, whose objects are the vertices V and whose morphisms from c to d are the paths from c to d. The reason for the importance of Set is that, if we generalize the definition of enriched category (Definition 2.46), we find that categories in the sense of Definition 3.6 are exactly Set-categories so categories are V-categories for a very special choice of V.
- 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.
