Label the following statements as true or false. (a) Any linear operator on an
Chapter 5, Problem 1(choose chapter or problem)
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.
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