Hi there, any help with this would be lovely, and full workingis greatly appreciated. thank you! :)
Let M = and consider the linear operator TM: R3 rightarrow R3 given by TM(v) = Mv. Show that the vectors a = (0 3 4), b = (-1 2 2), and c = (3 -1 1) are all eigenvectors of TM, and find the corresponding eigenvalues. For each real number mu show that these same vectors are eigenvectors for TM -mu I, and find the corresponding eigenvalues. Find a basis C of R3 for which the matrix of TM is diagonal. Find the matrix of (TM - 5I)-1(TM + I) with respect to the basis C. Find the transition matrix from the basis C to the standard basis of R3.
Show transcribed image text Let M = and consider the linear operator TM: R3 rightarrow R3 given by TM(v) = Mv. Show that the vectors a = (0 3 4), b = (-1 2 2), and c = (3 -1 1) are all eigenvectors of TM, and find the corresponding eigenvalues. For each real number mu show that these same vectors are eigenvectors for TM -mu I, and find the corresponding eigenvalues. Find a basis C of R3 for which the matrix of TM is diagonal. Find the matrix of (TM - 5I)-1(TM + I) with respect to the basis C. Find the transition matrix from the basis C to the standard basis of R3.