Math 51- Winter 2008 - Midterm Exam IIPlease circle the name of your TA:Zachary Cohn Jos´e Perea Nikola Penev Man Chun LiDaniel Mathews Theodora Bourni Anssi Lahtinen Isidora MilinCircle the time your TTh section meets: 10:00 11:00 1:15 2:15Your name (print):Student ID:Sign to indicate that you accept the honor code:Instructions: Circle your TA’s name and the time that you attend the TTh section. Readeach question carefully, and show all your work. You have 90 minutes to do all the problems.During the test, you may NOT use any notes, books, or calculators.Question 1 2 3 4 5 6 TotalMaximum 20 16 12 22 16 14 100ScoreProblem 1. (20 points total) Consider the matrix A =3 2−2 −2(a)(10 points) find the eigenvalues and the corresponding eigenvectors of A.2(b) (3pts) what are the eigenvalues of A99? (A is the matrix given above)(c) (3pts) is A99diagonalizable?(d) (4pts) if R is a region in R2of area 4, what is the area of its image under the lineartransformation with associated matrix A?3Problem 2. (16 pts total) Consider the linear transformation T that reflects vectors in R2across the line y = 2x.(a)(8 pts) find the matrix A corresponding to this linear transformation.(b)(8 pts) what are the eigenvalues and eigenvectors of A?4Problem 3. (12 points) Evaluate the following limit, or explain why the limit fails to exist.limx,y→0x2y2x4+ y4=5Problem 4. (22 points total) Assume you are standing at a point P of coordinates x = 20,y = 10 on a hillside whose height (in feet above sea level) is given byh(x, y) = 500 − x2+ 2xy + 3y2,where x points E (east), and y points N (north).(a) (6pts) suppose you start moving in the SW direction, do you ascent or descend?(b) (8pts) find the equation of the tangent plane to the graph of h(x, y) at the point P .(c) (8pts) use the result in (b) to approximate the change in height you experience if you movefrom P to the point of coordinates x = 21, y = 9.6Problem 5. (16 pts total) Consider A and B two 4 × 4 matrices with det A = 3 and det B = 2.Furthermore, denote by T : R4→ R4the linear transformation T (x) = Ax for all x ∈ R4.(a) (3 pts) find det(AB)(b) (3 pts) find det A−1(c) (3 pts) find det(2A)(e) (3pts) is the linear transformation T defined above invertible?(d) (4pts) can we find a basis B of R4in which T has matrix B?7Problem 6. (14 points) Assume x(t) is the position vector at time t of an ant moving smoothlyon a sphere of radius 5 centered at the origin. Prove that, at any moment, the velocity vectordxdtof the ant is perpendicular to its position
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