eigenvalues

Please do NOT send the answer back as an attachment. Please show work. Thank you. For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). Find the eigenvalues of T.

eigenvectors

Please do NOT send the answer back as an attachment. Please show work. Thank you. For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). Find the eigenvectors of T.

change of basis; eigenvectors

Please do NOT send the answer back as an attachment. Please show work. Thank you. For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). Write the change of basis matrix K from the basis F of R^3 which consists of the eigenvectors of T to the standard basis E for R^3.

diagonal matrix

Please do NOT send the answer back as an attachment. Please show work. Thank you. For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). The matrix A = [T]_E is similar to a diagonal matrix D = [T]_F. Write the diagonal matrix D, and demonstrate that it is indeed similar to A by producing the appropriate non-singular matrix and its inverse.

basis of a kernel

Please do NOT send the answer back as an attachment. Please show work. Thank you. For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z). What is a basis for the kernel of T?