Linear Algebra with Applications

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Unformatted text preview: i J Last Name: First Name: _ Mathematics 21b Second Midterm Examination November 22," 1999 Your Section (circle one): John Boiler Noam Elkies MWF 10 MWF 11 | No calculators are allowed. 1. Each of the following requires only a sentence or two of argument or explanation: lengthy computations and difficult proofs are not needed. 3 points each. (a) Let A be the matrix 3 1 4 1 5 9 —2 —6 —5 Show that the characteristic polynomial f A(/\) is NOT equal to either A4—3/\3+36/\2—90)\ or A3 —l—_3A2 + 36A m 90. (b) Show that, if A is any matrix of the form 5 0 0 0 0 *‘ 5' 0 0 0 >1: >1: ac * * a: * >1: a: a: >1: at: >t< '>|= * Where each =I< is a (possibly different) arbitrary number, then 5 is an eigenvalue of A of algebraic multiplicity at least 2. (c) Let A and B be the matrices 1 2 3 4 3 2 103 4 1 4 9 2 9 4 109 2 A _ 1 9 9 9 ’ B — 9 9 109 9 2 0 O 1 0 0 200 1 Giverl that det(A) = 186, show that det(B) = —18600. ((1) Let P be the parallelepiped in R3 defined by the vectors [El lél [ll What is the volume of P? What is the volume of T(P),I Where T is the linear transfor- mation with matrix 2 1 0 1 0 2 ? 0 2 1 2. i. (3 points) Define What it means for a matrix A to be orthogonal. ii. (3 points) Show that if A is orthogonal, so__is A1999. iii. (2 points) Is every orthogonal matrix invertible? Why? . Let P be the plane in IR3 Spanned by the column vectors 1 1 1 , 0 0 -—2 Let T be the orthogonal projection to P, i.e. T = projP. Note: You do mt need to find the matria: for T to solve any of the following. i) (3 points) Determine all the eigenvalues of T wee—+7.2: - ‘2’, banana-rm: ‘: ' - F”, @O‘Z‘ 8”“?5fi W‘égk wake-mac, i’E/«w-‘wwé’.’ éAlfiéwVa-é’w @éflhmgaéw-c maid? gram" _ r; J r -. ii) (3 points) Find an orthonormal basis for R3 consisting of eigenve‘ctors for T. iii) (4 points) Find the algebraic multiplicity of each eigenvalue of T. What is the character— istic polynomial of T? Is T invertible? Why or why not? 4( It is widely known that the Starship Enterprise is powered by Dilithiurn. Less well known is that of .each 10 tons of Dilithiurn, only 6 remain at the end of a year; of the remaining 4 tons, one is converted to energy to run the warp engines etc, but the other 3 transmute to 3 tons of Trilithium. Trilithium is not stable either: in a year, of each 10 tons Only 5 remain, with 3 tons converted to energy and the other 2 tons decaying back to Dilithium. At Stardatefl2000, the Starship’s reactor is loaded with 1000 tons of Dilithium. i) (2 points) Find a matrix A and an initial vector 22(0) that encode this discrete linear dynamical system, i.e. such that at Stardate 2000+ k, the Dilithium and Trilithiurn remaining in the reactor are the entries of the vector :c(k)"=<-rA-k3;0. ii) (3 points) Find the characteristic polynomial and eigenvalues of A, and compute a basis of eigenvectors. ' iii) (3 points) Compute a formula for the amounts of Dilithium and Trilithium remaining at Stardate 2000 + k as functions of It. What happens to these for large k? iv) (2 points) Many years and Star Trek episodes later, it is found that only one ton of Dilithium is left in the reactor. Approximately how much Trilithium is there at the same time? ...
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