 #  1. (a) Among all the points on the plane V in R4 spanned by t...

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 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.  (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.  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!)  (b) Determine the constituent matrices of A and use these to find a formula for An, for n = 1, 2,  3. (a) Write the linear difference equation 5an+2 - 4an+1 - an as a discrete linear system of the form Un+1 = Avn.  (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  (a) Find the eigenvalues of A and calculate their algebraic and geometric multi- plicities.  (b) Determine the Jordan canonical form J of A.  (c) Find an invertible matrix P such that P-1AP - J.  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  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  8. (a) Define what is meant by the orthogonal projection of a vector Rn onto a subspace VCR".  (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).  (c) State the Cayley-Hamilton Theorem and verify it for the matrix ^-(-12) A

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