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Unformatted text preview: Harvard University, Math 20 Fall 2010, Instructor: Rachel Epstein 1 Review sheet 2 1. Vector spaces, bases, and dimension (from supplement) (a) Know the definitions of vector space, basis, and dimension. Be able to identify when something is or is not a vector space, by the definition. (b) Be able to find a basis for a given vector space, and determine the dimension of the space. 2. Eigenvalues, eigenvectors, diagonalization (14.4-5): (a) Know the definition of eigenvalue and eigenvector, and how to use it. For example, be able to show that if ~v is an eigenvector of A with eigenvalue , then ~v is an eigenvector of A 2 with eigenvalue 2 . (b) Be able to find the characteristic polynomial, all real eigenvalues, eigenspaces, bases for eigenspaces, and algebraic and geometric multiplicities of eigen- values. Also be able to find an eigenbasis if one exists. (Note: I will not make you factor any polynomials of degree 3 or higher, but you might have to factor a quadratic polynomial to find eigenvalues, so you should be able to use the quadratic formula.) (c) Be able to find a diagonal matrix...
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