MATH 51 FINAL EXAM December 11, 2006Instructions:• No calculators, books, notes, or electronic devices may be used during the exam.• You have 3 hours.• There are 16 problems, each with multiple parts. Many questions have short answers requiringno computation. The point value of each part of each problem is indicated in brackets at thebeginning of that part. You should work quickly so as to not leave out problems towards theend of the exam.• Show computations on the exam sheet. If extra space is needed use the back of a page.Name:(print clearly)Signature:(for acce ptance of honor code)Your TA/discussion section (circle one):Antebi (15, 18)Ayala (3, 6)Easton (14, 17)Fernanadez (2, 5)Kim (8, 11)Koytcheff (9, 12)Lo (21, 24)Rosales (26, 27)Tzeng (20, 23)Zamfir (29, 30)Schultz (51A)Problem 1 (10 points)Problem 2 (10 points)Problem 3 (10 points)Problem 4 (10 points)Problem 5 (10 points)Problem 6 (10 points)Problem 7 (10 points)Problem 8 (10 points)Problem 9 (10 points)Problem 10 (10 points)Problem 11 (10 points)Problem 12 (10 points)Problem 13 (10 points)Problem 14 (10 points)Problem 15 (10 points)Problem 16 (10 points)Total (160 points)1. Suppose rref(A) =1 2 0 1 20 0 1 1 20 0 0 0 0and suppose you know that A1234−5=−179.(a) [5] Write in parametric form all solutions of the system of equations Ax =−179.(b) [5] Denote the i-th column of A by ai. Suppose a2=246and a4=1−1−1. Find A.[Hint: Make use of some linear dependence relations betwe en the columns of A.]12. For each of the following subsets S of R3determine if S is a subspace of R3. If not, give areason. If S is a subspace you don’t need to prove that, but give a basis of S.(a) [2] S =xyzx −2y + 3z = 2(b) [4] S =Allxyzorthogonal to both120and0−13(c) [4] S = span10−1,01−1,32−523. (a) [4] For which choice(s) of constant k is the matrix0 1 11 2 k1 4 k2not inve rtible?(b) [3] Let A =cos θ −sin θsin θ cos θ. Find det(A) and A−1.(c) [3] If B is an n × n matrix, find a formula for det(3B) in terms of det(B).34. Let Rθbe the linear transformation that rotates R3about the y-axis by θ radians in thedirection taking the positive x-axis toward the positive z-axis.(a) [4] Find the matrix for Rθwith respec t to the standard basis of R3.(b) [6] Compute A99where A =√3 0 −10 2 01 0√3.[Hint: Think geometrically. Note sin(π6) =12. What is12A?]45. As a reward for this problem, you will find an explicit formula for the Fibonacci sequence a0,a1, a2, a3, . . . defined recursive ly by a0= 0, a1= 1, an= an−1+ an−2(so the terms go 0, 1,1, 2, 3, 5, 8, 13, . . . ).(a) [3] Let xn=anan−1. Circle a matrix A so that Axn−1= xn.A =0 11 1A =0 −1−1 1A =1 11 0A =1 −1−1 0A =1 10 1A =1 −10 1(b) [2] Find the eigenvalues of A.λ = µ =5[Problem 5 continued] To correctly answer the remaining questions it is not necessary thatyou have correctly found λ and µ. You may assume that1−µis an eigenvector of A witheigenvalue λ, and1−λis an eigenvector for A with eigenvalue µ. Note that x1=a1a0=10.(c) [3] Use the diagonalization idea to solve for xnand circle the correct answer.i. xn= An−110= C−1Dn−1C10, C =1 1−µ −λ, D =µ 00 λii. xn= An−110= C−1Dn−1C10, C =1 1−µ −λ, D =λ 00 µiii. xn= An−110= CDn−1C−110, C =1 1−µ −λ, D =µ 00 λiv. xn= An−110= CDn−1C−110, C =1 1−µ −λ, D =λ 00 µ(d) [2] Find the inverse of C and circle the correct answer.C−1=1λ − µλ 1−µ −1C−1=1λ − µ−λ −1µ 1C−1=1λ − µ−1 −µ1 λThe punch line of this problem, obtained by combining parts (a)-(d), is the formula:an=1λ − µ(λn− µn).66. Let T1and T2be the linear transformations that are reflections in R3across the planes V1and V2respectively, where V1is given by the equation x + y − z = 0 and V2is given by theequation 2x − y + z = 0.(a) [1] Find normal vectors n1to V1and n2to V2. (They do not need to be unit vectors.)(b) [1] Verify that n1∈ V2and n2∈ V1.(c) [1] Two planes are said to be orthogonal if their normal vectors are orthogonal. Verifythat V1and V2are orthogonal.7[Problem 6 continued](d) [3] Find a nonzero vector n3∈ V1∩ V2.(e) [2] Find one basis B of R3consisting of three vectors that are simultaneously eigenvec-tors of both T1and T2. (Remember T1and T2are the reflections across V1and V2respectively.)(f) [2] Show that T1◦ T2= T2◦ T1.87. Let B be the orthonormal basis of R3given in standard coordinates byv1=1/32/32/3v2=2/31/3−2/3v1=2/3−2/31/3.Let V = span{v1, v2} and let P : R3→ R3be the orthogonal projection onto the plane V .(a) [3] Write down the matrix B for P with respect to the orthonormal basis B = {v1, v2, v3}.(b) [3] Write down the change of basis matrix C with Cej= vjwheree1=100, e2=010, e3=001is the standard basis of R3. Write down C−1. [Hint: no computation needed](c) [4] Find the matrix A for the projection P with respect to the standard basis of R3.98. (a) [3] Let V ⊂ Rnbe a subspace and let P : Rn→ V be the orthogonal projection. RegardP as a linear transformation Rn→ Rn. What real numbers are possible eigenvalues ofP?(b) [3] If T : Rn→ Rnis a linear transformation that satisfies T3= T, what real numbersare poss ible eigenvalues of T?(c) [4] Show that any orthonormal set of vectors {v1, v2, . . . , vk} in Rnmust be linearlyindependent. [Recall that orthonormal means vi· vi= 1 and vi· vj= 0 if i 6= j.]109. Let f : R2→ R be a differentiable function satisfying:f(5, 6) = 5 f(5, 6.2) = 6f(5.1, 6) = 6.05 f(5, 6.1) = 5.5f(5.01, 6) = 5.1005 f(5, 5.99) = 4.95f(5.001, 6) = 5.010005(a) [2] Use all of the above data to give the best value of the partial derivative fx(x, y) atthe point (x, y) = (5, 6).(b) [2] Use all of the above data to give the best value of the partial derivative fy(x, y) atthe point (x, y) = (5, 6).(c) [6] Give a linear approximation of the function f. Use your approximation to estimatef(6, 4).1110. (a) [5] Let f(x, y, z) = ax2+ by2+ cz2where a, b,