(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).

(3) Let V be a finite-dimensional vector space and TE L(V). Let A1,...,Am denote the distinct nonzero eigenvalues of T. Prove that T is diagonalizable if and only if dim E(A1,T) + ...+ dim E(m, T) = dim range T.

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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The linear transformation T : R2 → R2 with matrixA = cos θ − sin θsin θ cos θ rotates vectors in the x y-plane counterclockwise through an angle θ, where 0 ≤ θ <2π. By arguing geometrically, determine all valuesof θ for which A has real eigenvalues. Find the realeigenvalues and the corresponding eigenvectors.