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[6] 1. (a) Among all the points on the plane V in R4 spanned by the vectors vi= (1,1,1,1) and U2 = (-1,0,1,2), find the one that is closest to the vector y = (-5,3,2,6). Moreover, express this closest vector as a linear combination of V1 and U2. [4] (b) Consider the data points x -1.0 1.0 2.0 y -2.5 1.5 1.0 3.0 Use your results from part (a) to determine the straight line which, in the vertical least square sense, best fits all the data points. [5] 2. (a) Find the eigenvalues of the matrix A = and calculate their algebraic and geometric multiplicities. Moreover, deter- mine all the eigenvectors of A. Is A diagonable? (Justify your answer!) [5] (b) Determine the constituent matrices of A and use these to find a formula for An, for n = 1, 2, [3] 3. (a) Write the linear difference equation 5an+2 - 4an+1 - an as a discrete linear system of the form Un+1 = Avn. [7] (b) Solve the difference equation when ao = 2, a1 1, and use your solution to calculate lim an. n-00 Page 2 of 4 pages 4. A certain country has two provinces P1 and P2. The yearly censes taken in the country over the last 20 years shows that each year 10% of the residents of P1 move to P2, and that 15% of the residents of P2 move to P1. 3 (a) Summarize this information in a 2-state Markov chain Unt1 = Avn. Be sure to identify the states of the system and to write down the transition matrix A, and to explain what the entries of Vn mean. 2 (b) Assume that each province has 100,000 people today. How many people are expected be in each province 2 years from now? 5 (c) Verify that A is power convergent and find limno+ An. Assuming that the above trend continues, what is the population distribution of the country in the long run? 5. Consider the matrix A 2 [4] (a) Find the eigenvalues of A and calculate their algebraic and geometric multi- plicities. [2] (b) Determine the Jordan canonical form J of A. [4] (c) Find an invertible matrix P such that P-1AP - J. [10] 6. Find the characteristic polynomial and all the generalized geometric multiplicities VA(12) of the matrix -1 (GID) and use these to determine the Jordan Canonical Form J of A. Hint: Use the fact that -CMC and that A3=0.] Page 3 of 4 pages [10] 7. The matrix 3 2 21 3 1 2 3 has characteristic polynomial [You do not need to verify this fact. Show that A is power convergent, and calculate lim A". n-00 [2] 8. (a) Define what is meant by the orthogonal projection of a vector Rn onto a subspace VCR". [5] (b) Let A and B be two mxm matrices which are similar. Prove that An is similar to Bn, for all n 1, and that chA(t) = chB(t). [3] (c) State the Cayley-Hamilton Theorem and verify it for the matrix ^-(-12) A

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