CHEM 151B Midterm: Chem 151B UCSC Exam 2 2005
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You have 60 minutes to answer all four questions. Consider the matrix: find "s eigenvalues, find "s eigenvectors, argue that is diagonalizable and provide the matrices that accomplish its diagonal- ization. Hint for the last part: is diagonalizable if we can nd matrices and , with diagonal, such that 1 = . Show that if and are similar matrices then they have the same eingenvalues. Prove that any metric space that contains a nite numbr of elements is sequentially compact. Let be a linear transformation : . For any integer 1, de ne : as. Assume that there is an such that ( ) = 0 but 1( ) 6= 0 for some 0. ( ) 2( ) 1( ) are linearly independent.
