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1 1 (6) 1. Consider the matrix A = 0 1 0 . 0 -3 4 Find out whether A is diagonalizable and if so, find a diagonalization of A. 2 1 1 (7) 2. Consider the vectors = and the vector x= 1 3 2 2 3 Consider the linear subspace W = Span{a1,a2} of R4. (a) Determine an orthogonal basis of W. (b) Determine w € W and wit E w with x = w + wR. 2 5 (7) 3. Consider the real system X' = AX, with A = 2 0 and x(t) = [nd (a) Find the general real solution of this system. (b) Make a sketch of the phase portrait of this system. If appropriate, explain with a direction-arrow whether the motion is clockwise or counter clockwise. (c) Classify the critical point (0,0) as to type and determine whether it is stable, asymp- totic stable or unstable. (15) 4. Consider the heat conduction problem for the function 21(I., = un, with 0 < x < 2 en t > 0. We also have the following initial values and boundary values: (0, = 1. u(2,t) = 3, (a) Find the steady-state solution v(I) of the partial differential equation with the given boundary values. (b) Put y(z,t) = 21(I.t) - v(z). Write a partial differential equation for y(x,1 t). together with initial values and boundary values. (c) Use the method of separation of variables to solve y(x,t). (d) Solve u(x,t)

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