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18.06 Professor Johnson Quiz 2 April 1, 2009Grading123Total:Your PRINTED name is:Please circle your recitation:(R01) M2 2-314 Qian Lin(R02) M3 2-314 Qian Lin(R03) T11 2-251 Martina Balagovic(R04) T11 2-229 Inna Zakharevich(R05) T12 2-251 Martina Balagovic(R06) T12 2-090 Ben Harris(R07) T1 2-284 Roman Bezrukavnikov(R08) T1 2-310 Nick Rozenblyum(R09) T2 2-284 Roman Bezrukavnikov1 (20 pts.) (a) If P is the projection matrix onto the null space of A, then P y − y,for any y, is in the space of A.(b) If Ax = b has a solution x, then the closest vector to b in N(AT) is(best answer).(c) If the rows of A (an m×n matrix) are independent, then the dimensionof N(ATA) is .(d) If a matrix U has orthonormal rows, then I = and the pro-jection matrix onto the row space of U is . (Your answersshould be the simplest expressions involving U and UTonly.)2This page intentionally blank.32 (30 pts.) The matrixA =1 2 1 −72 4 1 −51 2 2 −16is converted to row-reduced echelon form by the usual row-elimination steps,resulting in the matrix:R =1 2 0 20 0 1 −90 0 0 0(♣) The minimum number of columns of A that form a dependent set ofvectors is . The maximum number of columns of A thatforms an independent set of vectors is .(♦) Give an orthonormal basis for the row space of A. (Careful: be sureyou start with a basis for the row space, not containing any dependentvectors.) Your answer may contain square roots left as√some number.(♠) Given the vector b =2 5 −9 3T, compute the closest vector pto b in the row space C(AT)? (Hint: less calculation is needed if youuse your answer from ♦.)(♥) In terms of your answer p to ♠ above, what is the closest vector tob in the nullspace N (A)? (No calculation required, and you need nothave solved ♠: you can leave your answer in terms of p and b.)4This page intentionally blank.53 (20 pts.) You are told that the least-square linear fit to three points (0, b1), (1, b2),and (2, b3) is C + Dt for C = 1 and D = −2. That is, the fit is 1 −2t.In this question, you will work backwards from this fit to reason about theunknown values b =b1b2b3Tat the coordinates t = 0, 1, 2.(i) Write down the explicit equations that b must satisfy for 1 − 2t to bethe least-square linear fit. (The points do not have to fall exactly onthe line.)(ii) If all the points fall exactly on the line 1 − 2t, then b = .Check that this satisfies your equations in (i).(iii) More generally, if all the points fall exactly on any straight line, thenb is in the space of what matrix? (Write down thematrix.)6This page intentionally


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MIT 18 06 - QUIZ - 18.06

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