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Math 51 Exam 2 — May 22, 2007Name :Section Leader: Theodora Peter Eric Henry Baosen(Circle one) Bourni Kim Schoenfeld Segerman Wu• 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.• You have 90 minutes. This is a closed-book, closed-notes exam. No calculators or other electronicaids will be permitted.• 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.• 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 code with respect to this examination.”Signature:The following boxes are strictly for grading purposes. Please do not mark.1 12 pts2 12 pts3 10 pts4 10 pts5 12 pts6 8 pts7 14 pts8 7 pts9 15 ptsTotal 100 ptsMath 51, Spring 2007 Exam 2 — May 22, 2007 Page 2 of 101. (12 points)(a) Find the eigenvalues of the following matrix.A =1 −1 11 −1 1−1 1 1(b) Compute the eigenspace of the largest eigenvalue you found in part (a).(c) Is there a nonzero vector−→v ∈ R3such that A−→v =−→v ? Briefly explain.Math 51, Spring 2007 Exam 2 — May 22, 2007 Page 3 of 102. (12 points) Let A =1 3 −34 −2 16.(a) Find a basis for the orthogonal complement of the null space N(A).(b) Given the matrix A above, fill in the blanks.dim(N(A)) + dim(N(A)⊥) =dim(C(A)) + dim(C(A)⊥) =dim(N(A)) + dim(C(A)) =(c) Using your answers to parts (a) and (b), give the dimensions of the four subspaces.dim(N(A)) = dim(N(A)⊥) =dim(C(A)) = dim(C(A)⊥) =Math 51, Spring 2007 Exam 2 — May 22, 2007 Page 4 of 103. (10 points) Let V be the 2-dimensional subspace of R3with basisB =12−1,110.(a) Express−→v =1−2Bin standard coordinates.(b) Express the following vector in B coordinates.−→w =23−1Math 51, Spring 2007 Exam 2 — May 22, 2007 Page 5 of 104. (10 points) Find the inverse of the given matrix.1 0 2−1 −2 −22 0 5Math 51, Spring 2007 Exam 2 — May 22, 2007 Page 6 of 105. (12 points) Let V be the 2-dimensional subspace of R3with basisB = {−→v1,−→v2} =122,443(a) Use the Gram-Schmidt process to find an orthonormal basis {−→w1,−→w2} for V .(b)−→v1and−→v2are sketched in V below. Draw the vector−→w2in the figure.−→v1−→v2(c) Find the distance from−→v2to the line spanned by−→v1.(You may use your work from parts (a) and (b)).Math 51, Spring 2007 Exam 2 — May 22, 2007 Page 7 of 106. (8 points)(a) Use the fact that det AT= det A to show that the determinant of an orthogonal matrix is ±1.(b) The volume of the unit sphere S in R3is43π. If T : R3→ R3is a linear transformation representedby orthogonal matrix, what is the volume of the image T (S)? Explain.Math 51, Spring 2007 Exam 2 — May 22, 2007 Page 8 of 107. (14 points) Mark each statement below as true or false by circling T or F. No justification is necessary.T F If A, B, and C are matrices such that AB = AC, then B = C.T F If A is a 2 × 2 matrix with det(A) = 0, then one column is a multiple of the other.T F If A and B are n × n matrices with det(A) = 2 and det(B) = 3, then det(A + B) = 5.T F Suppose that A is an n×n matrix with rref(A) = In. It follows that A is diagonalizable.T F The components of a function f : R → Rmare vectors.T F lim(x,y)→(0,0)x2y3− 2xyxy= −2T F The function f(x, y) =(y4−x4x2+y2when(x, y) 6= (0, 0)2 when(x, y) = (0, 0)is continuous.Math 51, Spring 2007 Exam 2 — May 22, 2007 Page 9 of 108. (7 points) Match the graph of the surface with one of the contour maps.Write the letter of the contour map corresponding to the graph of the given function in the box:f(x, y) = x2+y24f(x, y) = e1−x2+y2f(x, y) = e1−x2−y2f(x, y) = ln |y − x2|Math 51, Spring 2007 Exam 2 — May 22, 2007 Page 10 of 109. (15 points) Consider the functionf(x, y) = ln(x2+ y2− 4x + 6y + 15).(a) Find the points (a, b) where fx(a, b) = 0 and fy(a, b) = 0 simultaneously.(b) For each of the points you found in part (a), determine the equation of the tangent plane to thegraph of f at that point.(c) For the function F (x, y, z) = (ln(x2+ y2− 4x + 6y + 15), xyz, x2z + zexy), compute DF


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Stanford MATH 51 - Study Guide

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