7. Let T : R2 - R2 be a linear transformation given by Find a basis of eigenvectors B for R2, and find the matrix [T],. What do you notice about [T]?? 8. Let V = span{er, e ~}, and let D : V - V be differentiation with respect to x. Another basis of V is C = { cosh (x), sinh(x) }, where cosh(x) = sinh(x) = 2 2 (a) Find the matrix of D relative to C, [D]C. (b) Find the eigenvalues 1, 12 of [D]c and their corresponding eigenvectors u = U2 V = (c) Interpreting u and v from part (b) as component vectors with respect to the basis C, we get u(x) = uj cosh(x) + u2 sinh(x) and v(x) = v1 cosh(x) + v2 sinh(x), so that u, v E V. Show by direct computation that u and v are eigenvectors of D. 9. In each case V is a vector space, T : V - V is a linear transformation, and v is a vector in V. Determine whether v is an eigenvector of T. If so, give the associated eigenvalue. (a) T : P2(R) -+ P2(R), where T(ao + aix + azz2) = (a2 -do) + 2aix + (ao + a1 + a2)22 and v = 1+ 2x + 3x2. (b) T : M2(R) + M2(R), where a + 26 2a + b -ctd 2d , v= o o