Let X be a binomial random variable as in Example 33.5. Prove that E.X / D np.

M303 Section 5.1 Notes- Introduction to “Eigenstuff”; Diagonalization 11-30-16 From here on, only consider square matrices ( ) If is × , then gives rise to liner map :ℝ → ℝ givn by = o Since source and target dimensions are identical, can compare inputs and outputs Focus on vectors on which has particularly simple effect Eigenvector of matrix - nonzero vector such that = , where ℝ o uniquely determined and called the eigenvalue of (for ); we call a -eigenvector o Eigenvalues of are scalars for which there exist -eigenvectors may not have any eigenvalues or eigenvectors E-vectors nonzero by definition ( = , would not be unique), but e-values can be 0 Ex. Let = [ −2], = (−1,1), and = (2,1). Is either or an e-vector of 1 0 o = [3 −2 ][ ] 1 0 1 = [−3 − 2 ] −1 − 0 = [−5 ] −1 ≠ → not e-vector o = [3 −2 2][ ] 1 0 1 = [6 − 2] 2 − 0 = [ ] 2 = 2 → is e-vector with e-value 2 1 6 Ex.