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Unformatted text preview: MA 527, Fall 2009, Purdue University Sample questions for exam 1 with solutions This is a collection of problems that are similar to problems that you will find on the actual exam. The actual exam will be inclass and contain a mix of computation, true/false or multiple choice, and proof. You will be allowed one standard page of notes (both sides), but aside from that the exam will be closed book and notes, no calculator or other computational aids. Chapter 7: Basic linear algebra: Rank, basis, independence, solving systems, determinants, inverse. Chapter 8: Eigenvalues/vectors: Finding eigenvalues/vectors/basis for eigenspace diagonalization/similarity properties Chapter 4: Systems of differential equations: Solution of systems  homogeneous and nonhomogeneous, critical points, types of crit ical points/stability Sample problems 1. True/False. No explanation necessary. Note that in mathematics, if a statement is sometimes true and sometimes false, then it is false . (a) If A is diagonalizable, then its eigenvalues are distinct. False. E.g. the matrix I is diagonal, hence diagonalizable, but all eigenvalues are 1. (b) If all the eigenvalues of A equal 2, then there is a matrix P so that P 1 AP = 2 I . False. E.g., the matrix 2 1 0 2 has both eigenvalues 2 but is defective, hence not diagonalizable. (c) If B = C 1 AC , then B 3 = C 1 A 3 C . True. Expand B 3 using the definition given, then cancel the terms CC 1 . (d) If A 1 exists, then all the eigenvalues of A are nonzero....
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 Spring '08
 WEITSMAN
 Advanced Math

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