Label the following statements as true or false. (a) Any linear operator on an n-dimensional vector space that has fewer than n distinct eigenvalues is not diagonalizable. (b) Two distinct eigenvectors corresponding to the same eigenvalue are always linearly dependent. (c) If A is an eigenvalue of a linear operator T, then each vector in EA is an eigenvector of T. (d) If Ai and A2 are distinct eigenvalues of a linear operator T, then EAlnEA2 = {0}. (e) Let A G Mnxn (F) and 0 = {vi,v2, ,vn} be an ordered basis for F n consisting of eigenvectors of A. If Q is the nxn matrix whose jth column is Vj (1 < j < n), then Q~lAQ is a diagonal matrix. (f) A linear operator T on a finite-dimensional vector space is diagonalizable if and only if the multiplicity of each eigenvalue A equals the dimension of EA(g) Every diagonalizable linear operator on a nonzero vector space has at least one eigenvalue. The following two items relate to the optional subsection on direct sums. (h) If a vector space is the direct sum of subspaces Wl5 W2 ,... , \Nk, then Wi n Wj = {0} for 1\ j. (i) ^ V = W i i=l then V = Wi W2 CE and WiDWj = {tf} for i ^ j, Wfc.

HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Homogeneous Equations David Levermore Department of Mathematics University of Maryland 12 March 2012 Because the presentation of this material in lecture will diﬀer from that in the book, I felt that notes that closely follow the lecture presentation might be appreciated. 1. Introduction 1.1. Normal Form and Solutions 2 1.2. Initial Value Problems 2 1.3. Intervals of Deﬁnition 4 2. Homogeneous