(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).

What is the eigenvalue of the operator T, given by T(x) = 3x + 2, for the vector x = [1, 2]?

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Let - Q(x) = 35x²7x27x3 + 18x1x2 + 18x1x3 − 18x2x3. Find a unit vector x in R³ whose first entry is positive at which Q(x) is minimized, subject to xx = 1. [Hint: The eigenvalues of the matrix of the quadratic form Q are 2, -19, and 38.]

(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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