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Understanding Linear Mappings, Eigenvalues, and Eigenvectors in Linear Algebra

This educational content explores core concepts of linear algebra, focusing on linear mappings and their properties, such as the dimension of null spaces. It discusses the relationship between linear transformations and their ranges, providing examples to illustrate how pre-images correspond to cosets of null spaces. Additionally, it addresses the similarity of matrices, the characteristics of eigenvalues and eigenvectors, and presents various examples, including those that exhibit specific relationships to null spaces. This comprehensive overview is ideal for students learning linear algebra.

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Understanding Linear Mappings, Eigenvalues, and Eigenvectors in Linear Algebra

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  1. 1.)

  2. 2.) If T is a linear mapping such that the dimension of the null space of T is greater than zero, then for every vector in the range of T the pre-image of is a coset of the null space of T. Give an example to illustrate this concept.

  3. 3.) If A is similar to B , is A necessarily similar to AB? Give reasons for your answer – either a proof or a counterexample.

  4. 4.) Find eigenvalues and eigenvectors for A , and if possible a diagonal matrix that is similar to A.

  5. 5.) Find eigenvalues and eigenvectors for each of the following:

  6. 6.) Give an example of a matrix A such that:

  7. 7.) M= Give at least one eigenvector that belongs to the null space of M. Give at least one eigenvector that does NOT belong to the null space of M. Give the corresponding eigenvalue.

  8. 8.) A is an nn matrix. aik represents the element in row i column k. aik = (-1)i + k for all i and k. Give an example of an eigenvalue that is different from 0 and describe an eigenvector to go with it.

  9. 9.)

  10. 60° 10.) The vectors and form a 60 degree angle. b = ?

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