# One-to-one. Definition of kernel: Let be a linear transformation. The kernel of L,, is the

 Дата канвертавання 22.04.2016 Памер 24.97 Kb.
 Kernel and Range Definition of one-to-one transformation: A linear transformation is said to be one-to-one if for all in V, implies that (or implies ). Example: . Is L one-to-one? [solution:] Let . Then, . . Therefore, L is one-to-one. Definition of kernel: Let be a linear transformation. The kernel of L, , is the subset of V consisting of all vectors such that . Example: Let . What is ? [solution:] is the set consisting of all vectors such that . That is, is the solution space of . Important result: Let be a linear transformation. Then, is a subspace of V. [proof:] For any , . Then, 1. . 2. . By 1, 2, is a subspace of V. Important result: A linear transformation is one-to-one if and only if [proof:] Since L is one-to-one, implies there is only one vector 0 such that . . Suppose . Then, . Since , that implies . Therefore, L is one-to-one. Important result: Let be a linear transformation defined by , where A is a matrix. Then, the following conditions are equivalent: . has only the trivial solution. has a unique solution for every . is one-to-one. Definition of range: Let be a linear transformation. The range of L, , is the subset of W consisting of all vectors that are images of vectors in V (i.e., for any , there exists in V such that ). If , L is said to be onto. Important result: Let be a linear transformation. Then, is a subspace of V. [proof:] Since , . Thus, is not a empty set. For any , there exist in V such that . Then, 1. . Then, since . 2. . Then, since By 1, 2, is a subspace of V. Example: Let , . Is L onto? What is range L? [solution:] Suppose L is onto. Then for any vector , there exists such that . That is span is a basis for . But . Thus, is not a basis and it is a contradiction. Therefore, L is not on-to. (b) Note: As the linear transformation defined by then and . Important result: Let be a linear transformation of an n-dimensional vector space V into a vector space W, then, , where is the dimension of some vector space. Note: As the linear transformation defined by

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