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- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/09%3A_Vector_Spaces/9.07%3A_Isomorphisms\[\begin{aligned} T \left( k \left[a1b1c1d1\right] + p \left[a2b2c2d2\right] \right) &= T \left( \left[\begin{ar...\[\begin{aligned} T \left( k \left[a1b1c1d1\right] + p \left[a2b2c2d2\right] \right) &= T \left( \left[ka1kb1kc1kd1\right] + \left[pa2pb2pc2pd2\right] \right) \\ &= T \left( \left[ka1+pa2kb1+pb2kc1+pc2kd1+pd2\right] \right) \\ &= \left[ \begin{array}{c…
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_(Sklar)/03%3A_Homomorphisms_and_Isomorphisms/3.02%3A_Definitions_of_Homomorphisms_and_IsomorphismsIntuitively, you can think of a homomorphism as a “structure-preserving” map: if you multiply and then apply homormorphism, you get the same result as when you first apply homomorphism and then multip...Intuitively, you can think of a homomorphism as a “structure-preserving” map: if you multiply and then apply homormorphism, you get the same result as when you first apply homomorphism and then multiply. Isomorphisms, then, are both structure-preserving and cardinality-preserving. Homomorphisms from a group G to itself are called endomorphisms, and isomorphisms from a group to itself are called automorphisms.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/05%3A_Graph_Theory
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.05%3A_One-to-One_and_Onto_TransformationsThis section is devoted to studying two important characterizations of linear transformations, called One to One and Onto.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.11%3A_The_Matrix_of_a_Linear_TransformationYou may recall from Rn that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary v...You may recall from Rn that the matrix of a linear transformation depends on the bases chosen. This concept is explored in this section, where the linear transformation now maps from one arbitrary vector space to another.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/05%3A_Linear_Transformations/5.06%3A_Isomorphisms/5.6E%3A_Exercises_for_Section_5.6This page contains exercises on linear transformations and isomorphisms in vector spaces, focusing on defining transformations from R3 and R2. It covers properties of isomo...This page contains exercises on linear transformations and isomorphisms in vector spaces, focusing on defining transformations from R3 and R2. It covers properties of isomorphisms, proving conditions for transformations, exploring matrix representations, and finding inverses. The content also discusses constructing matrices that uphold the structure of transformations and their inverses, particularly regarding spans of vectors in higher dimensions.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/09%3A_Vector_Spaces/9.07%3A_Isomorphisms\[\begin{aligned} T \left( k \left[a1b1c1d1\right] + p \left[a2b2c2d2\right] \right) &= T \left( \left[\begin{ar...\[\begin{aligned} T \left( k \left[a1b1c1d1\right] + p \left[a2b2c2d2\right] \right) &= T \left( \left[ka1kb1kc1kd1\right] + \left[pa2pb2pc2pd2\right] \right) \\ &= T \left( \left[ka1+pa2kb1+pb2kc1+pc2kd1+pd2\right] \right) \\ &= \left[ \begin{array}{c…
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/05%3A_Vector_Spaces/5.07%3A_Isomorphisms\[\begin{aligned} T \left( k \left[a1b1c1d1\right] + p \left[a2b2c2d2\right] \right) &= T \left( \left[\begin{ar...\[\begin{aligned} T \left( k \left[a1b1c1d1\right] + p \left[a2b2c2d2\right] \right) &= T \left( \left[ka1kb1kc1kd1\right] + \left[pa2pb2pc2pd2\right] \right) \\ &= T \left( \left[ka1+pa2kb1+pb2kc1+pc2kd1+pd2\right] \right) \\ &= \left[ \begin{array}{c…
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Groups_and_Geometries_(Lyons)/02%3A_Groups/2.04%3A_Group_homomorphismsLet K be a subgroup of a group G. The set G/K of cosets of K forms a group, called a quotient group (or factor group), under the operation Let G be a group, let a be an elem...Let K be a subgroup of a group G. The set G/K of cosets of K forms a group, called a quotient group (or factor group), under the operation Let G be a group, let a be an element of G, and let Ca:G→G be given by Ca(g)=aga−1. The map Ca is called conjugation by the element a and the elements g,aga−1 are said to be conjugate to one another.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/08%3A_An_Introduction_to_Rings/8.02%3A_Ring_HomomorphismsIt turns out that ϕ is a ring homomorphism, where ker(ϕ) is the set of polynomials with 0 constant term. Suppose ϕ:R→S is a ring homomorphism such that R is a ring with 1,...It turns out that ϕ is a ring homomorphism, where ker(ϕ) is the set of polynomials with 0 constant term. Suppose ϕ:R→S is a ring homomorphism such that R is a ring with 1, call it 1R. Can you think of an example of a ring homomorphism where S has a multiplicative identity that is not equal to ϕ(1R)?
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/05%3A_Linear_Transformations/5.06%3A_IsomorphismsA mapping T:V→W is called a linear transformation or linear map if it preserves the algebraic operations of addition and scalar multiplication.