(C).(3%) Show that the set{1+x+x²+x³，1-x-x²+x³,1+x-x²- x³，1-x+x²-x³} can be used as a basis for P₃.

(B).(3%) When will the transformnation matrix have non-zero eigenvalues? Please explain.

(C). (4%) Is the transformation matrix similar to the identity matrix? Why?

(A).(3%) Give an example of two vectors in P3 that are orthogonal to each other.

(B).(3%) Give an example of a unit-length vector in P3.

(D).(3%) Use the set {1, x, x2, x3} as the basis for P3. Find the matrix representation of the linear transformation that does differentiation on polynomials in P3.

(E). (3%) Continue with part (D). Can the matrix representing differentiation be diagonalized? Why?

(A).(5%) (x + 3)2y"' -8(x+ 3)y'+ 14y =0

(B).(5%) x2y" - 4xy'+ 6y = Inx2

六、(10%,計算題) Find the solution y(x) of the following initial value problem for x ＞ 1. x2y"-2xy'+2y=xlnx, y(1)=1, y'(1)=0

(A). (5%) Consider a system of linear differential equations in the following vector form: Here v(t) is an n-dimensional column vector and M is an n x n matrix. Assume that the matrix M can be diagonalized by a similarity transformation:Here S is the transformation matrix and D is a diagonal matrix with λ1,λ2,... ,λn as its diagonal elements. The initial values of the equations form the vector v(0) and P(t) is a diagonal matrix with exp (λ1t),exp(λ2t), ,exp (λnt) as its diagonal elements. Please express the solution v(t) for t＞ 0 in terms of v(0), S and P(t) and explain why.