Search
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/05%3A_Linear_Transformations/5.07%3A_The_Kernel_and_Image_of_A_Linear_MapIn this section we will consider the case where the linear transformation is not necessarily an isomorphism.
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/05%3A_Vector_Spaces/5.08%3A_The_Kernel_and_Image_of_a_Linear_MapHere we consider the case where the linear map is not necessarily an isomorphism. First here is a definition of what is meant by the image and kernel of a linear transformation.
- 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/6.10.02%3A_Kernel_and_Image_of_a_Linear_TransformationTo prove part (a), note that a matrix A lies in ker P just when 0=P(A)=A−AT, and this occurs if and only if A=AT—that is, A is symmetric. of V with the pro...To prove part (a), note that a matrix A lies in ker P just when 0=P(A)=A−AT, and this occurs if and only if A=AT—that is, A is symmetric. of V with the property that {er+1,…,en} is a basis of ker T and {T(e1),…,T(er)} is a basis of imT.
- https://math.libretexts.org/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/05%3A_Linear_Transformations/5.08%3A_The_General_Solution_of_a_Linear_SystemIt turns out that we can use linear transformations to solve linear systems of equations.
- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/07%3A_Vector_Spaces/7.07%3A_Sums_and_Intersections/7.7E%3A_Exercises_for_Section_7.7This page discusses exercises on vector spaces and subspaces in Rn, focusing on geometric interpretations, dimensions of sums and intersections of subspaces U and W, and specific...This page discusses exercises on vector spaces and subspaces in Rn, focusing on geometric interpretations, dimensions of sums and intersections of subspaces U and W, and specific examples of bases. It clarifies that U+W may span the entire space or have distinct dimensions, while intersections can vary in dimension.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/07%3A_Linear_Transformations/7.02%3A_Kernel_and_Image_of_a_Linear_TransformationTo prove part (a), note that a matrix A lies in ker P just when 0=P(A)=A−AT, and this occurs if and only if A=AT—that is, A is symmetric. of V with the pro...To prove part (a), note that a matrix A lies in ker P just when 0=P(A)=A−AT, and this occurs if and only if A=AT—that is, A is symmetric. of V with the property that {er+1,…,en} is a basis of ker T and {T(e1),…,T(er)} is a basis of imT.
- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/05%3A_Linear_Transformations/5.09%3A_The_General_Solution_of_a_Linear_SystemIt turns out that we can use linear transformations to solve linear systems of equations.
- https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/04%3A_Linear_Transformations/4.07%3A_The_Kernel_and_Image_of_A_Linear_MapIn this section we will consider the case where the linear transformation is not necessarily an isomorphism.
- https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/05%3A_Vector_Spaces_and_Subspaces/5.10%3A_The_Kernel_and_Image_of_a_Linear_MapHere we consider the case where the linear map is not necessarily an isomorphism. First here is a definition of what is meant by the image and kernel of a linear transformation.
- https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/07%3A_Homomorphisms_and_the_Isomorphism_Theorems/7.01%3A_HomomorphismsThe next theorem tells us that under a homomorphism, the order of the image of an element must divide the order of the pre-image of that element. The next theorem tells us that two elements in the dom...The next theorem tells us that under a homomorphism, the order of the image of an element must divide the order of the pre-image of that element. The next theorem tells us that two elements in the domain of a group homomorphism map to the same element in the codomain if and only if they are in the same coset of the kernel.
- https://math.libretexts.org/Bookshelves/Linear_Algebra/A_First_Course_in_Linear_Algebra_(Kuttler)/09%3A_Vector_Spaces/9.08%3A_The_Kernel_and_Image_of_a_Linear_MapHere we consider the case where the linear map is not necessarily an isomorphism. First here is a definition of what is meant by the image and kernel of a linear transformation.