(19 1. Let T V V be a linear operator on the vector space V over Q that has characteristic polynomial XT(x) = -(x-1)4(x-2)3(x+1)2 and has minimal polynomial (r - 1)2(x-2)²(x + 1)². Find the possible Jordan normal forms of T. (Arrange the Jordan blocks so that the eigenvalues appear on the diagonal in a non-increasing sequence.)

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

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Consider the vectors x(t) = ( and x2) (t) = 6. 4t (a) Compute the Wronskian of x) and x2). W (b) In what intervals are x and x2) linearly independent? x(1) D = (c) What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by x and x2)? x(1) Choose one ▼ of the coefficients of the ODE in standard form must be discontinuous at to : (d) Find the system of equations x' = P(t)x.

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5. Let T: V→ V be a linear operator on a finite-dimensional F-vector space V, and let f(t) be the characteristic polynomial of T. Show that T is diagonalizable if and only if f(t) splits over F and rank(T – XI) = rank((T – XI)²) for every eigenvalue A of T. (You can use the theorem that T has a Jordan basis iff f(t) splits, and the theorem that says nullity ((T – XI)²) – nullity(T – XI) =···)

2. Let T = -2 1 1 2-3 1 0 0 3 and denote 13 = 1 0 0 1 0 0 001 (a) i). Calculate det (T-xI3) for any r to find the characteristic polynomial of T; ii). Determine the eigenvalues of T. (b) Find an eigenvector of T associated with each of the eigenvalues of T. a (c) Express an arbitrary vector v = b in R³, where a, b, c ER, as a linear combi- C nation of eigenvectors of T.