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18.06 Professor Johnson Quiz 1 March 2, 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.) Your classmate, Nyarlathotep, performed the usual elimination steps toconvert A to echelon form U, obtaining:U =1 4 −1 30 2 2 −60 0 0 0.(a) Find a set of vectors spanning the nullspace N(A).(b) If Uy =9−120, find the complete solution y (i.e. describe all possiblesolutions y).(c) Nyarla gave you a matrixL =1 0 02 1 0−1 3 1and told you that A = LU. Describe the complete sequence of elimi-nation steps that Nyarla performed, assuming that she did eliminationin the usual way starting with the first column and eliminating down-wards. That is, Nyarla first subtracted times the first row fromthe second row, then subtracted times the first row from thethird row, then subtracted .(Be careful about signs: adding a multiple of a row is the same as sub-tracting a negative multiple of that row.)(d) If Ax =026, then Ux = .2This page intentionally blank.32 (20 pts.) Which of the following (if any) are subspaces? For any that are not asubspace, give an example of how they violate a property of subspaces.(I) Given some 3 × 5 matrix A with full row rank, the set of all solutionsto Ax =111.(II) All vectors x with xTy = 0 and xTz = 0 for some given vectors y andz.(III) All 3 × 5 matrices with123in their column space.(IV) All 5 × 3 matrices with123in their nullspace.(V) All vectors x with kx − yk = kyk for some given fixed vector y 6= 0.4This page intentionally blank.53 (20 pts.) A is a matrix with a nullspace N(A) spanned by the following three vectors:12−13,0114,−1−131.(α) Give a matrix B such that its column space C(B) is the same as N(A).(There is more than one correct answer.) [Thus, any vector y in thenullspace of A satisfies Bu = y for some u.](β) Give a different possible answer to (α): another B with C(B) = N(A).(γ) For some vector b, you are told that a particular solution to Ax = bisxp=1234.Now, your classmate Zarkon tells you that a second solution is:xZ=1130,while your other classmate Hastur tells you “No, Zarkon’s solutioncan’t be right, but here’s a second solution that is correct:”xH=1131.Is Zarkon’s solution correct, or Hastur’s solution, or are both correct?(Hint: what should be true of x − xpif x is a valid solution?)6This page intentionally


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MIT 18 06 - Quiz 1

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