Math 51 Exam 1 — April 22, 2008Name :Section Leader: Fai Joseph David Anca Bezirgen(Circle one) Chandee Cheng Fernandez-Duque Vacarescu VeliyevSection Time: 10:00 11:00 1:15 2:15(Circle one)• Complete the following problems. You may use any result from class you like, but if you cite atheorem be sure to verify the hypotheses are satisfied.• In order to receive full credit, please show all of your work and justify your answers. You do not needto simplify your answers unless specifically instructed to do so.• If you need extra room, use the back sides of each page. If you must use extra paper, make sure towrite your name on it and attach it to this exam. Do not unstaple or detach pages from this exam.• You have 90 minutes. This is a closed-book, closed-notes exam. No calculators or other electronicaids will be permitted. If you finish early, you must hand your exam paper to a member of teachingstaff.• It is your responsibility to arrange to pick up your graded exam paper from your section leader ina timely manner. You have only until Tuesday, May 6, to resubmit your exam for any regradeconsiderations; consult your sec tion leader about the exact details of the submission process.• Please sign the following:“On my honor, I have neither given nor received any aid on thisexamination. I have furthermore abided by all other aspects of thehonor c ode with respect to this examination.”Signature:The following boxes are strictly for grading purposes. Please do not mark.1 10 5 142 15 6 153 14 7 124 20 Total 100Math 51, Spring 2008 Exam 1 — April 22, 2008 Page 2 of 91. (10 points) C ompute, showing all steps, the reduced row echelon form of the matrix0 2 −1 11 1 2 41 1 3 51 1 1 3.Math 51, Spring 2008 Exam 1 — April 22, 2008 Page 3 of 92. (15 points) Consider the matrix A below, and its reduced row echelon form (which you do not haveto verify!):A =1 0 1 31 0 0 21 0 −1 11 0 1 3, rref(A) =1 0 0 20 0 1 10 0 0 00 0 0 0.(a) Find a basis for the null space of A. You do not need to prove that your collection is a basis.(b) Are the columns of A linearly independent? If so, explain why; if not, write a non-trivial linearrelation they s atisfy.Math 51, Spring 2008 Exam 1 — April 22, 2008 Page 4 of 9(ctd. from previous page) A =1 0 1 31 0 0 21 0 −1 11 0 1 3, rref(A) =1 0 0 20 0 1 10 0 0 00 0 0 0(c) Find a basis for the column space of A. You do not need to prove that your collection is a basis.(d) Give a specific vector b ∈ R4such that the equation Ax = b has exactly one solution x ∈ R4(and give this solution), or state why such a b does not exist.Math 51, Spring 2008 Exam 1 — April 22, 2008 Page 5 of 93. (14 points) In each of the parts below, a set of vectors in R3is specified. In each case, find anexpression for this set in parametric form, using linearly independent vectors. (Alternatively, ifapplicable, you may find a basis for the set.) Show the steps of your computations.(a) S =x =x1x2x3∈ R3x1+ x2+ x3= 1 and2x1− x2+ x3= −1.(b) Y is the set of all vectors in R3orthogonal to213. (Hint: you can approach this by solving asystem.)Math 51, Spring 2008 Exam 1 — April 22, 2008 Page 6 of 94. (20 points) Mark e ach statement below as true or false by circling T or F. No justification is necessary.T F For any ve ctors v1, v2, . . . , vkin Rk+1, the set {v1, v2, . . . , vk} is linearly independent.T F For any ve ctors v1, v2, . . . , vkin Rk+1, the set {0, v1, v2, . . . , vk} is linearly dependent.T F If the system Ax = b has infinitely many solutions, then there is at least one free variable.T F The null space of1 2 01 2 01 2 0is span2−10.T F If B is a 3-by-3 matrix such that the linear transformation T (x) = Bx defines a projectionof x ∈ R3onto one of the coordinate axe s of R3, then the column s pace of B hasdimension 1.T F The function T : R2→ R2defined by Tx1x2=x1+ 22x2is a linear transformation.T F If Q is any square matrix, then the null space of Q has dimension equal to the number ofzero rows in the reduced row echelon form of Q.T F There exists a subspace of R6of dimension 5.T F The matrix1 01 0is in reduced row echelon form.T F For any ve ctors x, y in Rn, ||x + y|| = ||x|| + ||y||.Math 51, Spring 2008 Exam 1 — April 22, 2008 Page 7 of 95. (14 points) Le t P be the plane in R3containing the points (−1, 0, 0), (0, 2, 0), and (0, 0, 3).(a) Find an equation of the plane P; give your answer as a single linear equation involving coordi-nates x, y, and z.(b) Find values (a, b, c) satisfying both of the following conditions simultaneously:• the point M = (a, b, c) lies on the plane P, and• the vector v =abcis normal (perpendicular) to the plane P.Math 51, Spring 2008 Exam 1 — April 22, 2008 Page 8 of 96. (15 points) Le t T : R4→ R6be the function defined by: Tx1x2x3x4=x3x2x4x300.(a) Is T a linear transformation? If so, give the matrix A associated to T ; if not, explain why.(b) Determine all vec tors x in R4such that T (x) = 0.(c) Let W be the set of vectors in R6which can be expressed as T (x) for an appropriate x ∈ R4.It is a fact that for this T , the set W is a subspace of R6. Find a set of vectors that spans W .Math 51, Spring 2008 Exam 1 — April 22, 2008 Page 9 of 97. (12 points)(a) Complete the sentence: A set V of vectors in Rnis a subspace if(b) If S is a set of vectors in Rn, let S⊥be the set of vectors v in Rnsatisfying v · w = 0 for everyw in S. Show that S⊥is a subspace of