midterm 2 fall 2001

# Linear Algebra with Applications

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Unformatted text preview: MATH 21b FALL 2001 EXAMINATION 2 Tuesday, 4 December, 2001. Name: Teaching Fellow (PLEASE CIRCLE) William Dale Spiro Stein Winter Karigiannis MWF 10am MWF 11am TuTh 103m Instructions: 1. Do not open this test until told to do so. 2. Please do not detach any pages from this exam. 3. You may use your calculator and one (1) page of notes not exceeding 8.5 by 11 inches In me. 4. You may use the backs of test sheets for scratch paper. or to continue your working on problems. If you write on the backs of the test sheets, please label your working very clearly. 5. SHOW ALL YOUR WORK. 6. Exam proctors are not permitted to answer questions regarding the content of the test. 7. Many of the questions have preciser worded instructions. Be sure to read all instructions carefully, and do all that is asked. 8. Please try your best - try to relax and Show us what you can do! 1. (13 points total) 1: Thcsctofvectors: W: y :2x—y+4z=0 isasubspaceofR’. (Thereisnonead z for you to demonstrate this fact.) (a) (4 points) Find a basis for W. (b) (4 points) Construct an orthononnal basis for W. (e) (5 points) Calculate the shortest distance from the point (I, l, l) to W. 2. (13 points total) In this pmblem, the matrix A will always be: DN-h- luau: l A=0 0 (a) {3 points) Find the characteristic polynomial of the matrix A. (b) (4 points) Is the mail-ix A diagonalizable or not? Explain how you can tell. (c) (6 points) The graphs shown below are the graphs of the characteristic polynomials of three by three manices. For each. decide if the menu is deﬁnitely diagonalizable. possibly ydiagonalizable or deﬁnitely not diagonalizable. 3. (13 points total) In this problem the matrix A will always refer to the 2 by 2 matrix: A = . 2 l l (a) (3 points) Show that the vector F1=[2] is an eigenvector of A and ﬁnd the corresponding eigenvalue. 1 (b) (3 points) Show that the vector 9’2 =|:1] is an eigenvector of A and ﬁnd the corresponding eigenvalue. (1:) (7 points) Suppose that the matrixA is the matrix for a discrete dynamical system: it!) = [:8] 20 +1) = A m) i(0)=[::1. Find an explicit formula for the solution vector 3:0). 4. (16 points total) In this quaition, you will always be dealing with two by two matrices. You are expected to use the two deﬁnitions given below to answer this question. Deﬁnition 1: A 2 by 2 rotation matrix is a matrix of the form: [cos(s) 4111(9)]- sin(9) cos(9) Deﬁnition 2: An outing-anal mtrtx is a square matrix whose columns are onhomnnal vectors. (a) (4 points) Show that a 2 by 2 rotation matrix is an orthogonal matrix. (b) (6 points) Show that the inverse of a rotation matrix is equal to the transpose of the rotation matrix. Note: sint-é?) = —sin(6) and cone-9) = mum. (c) (6 points) Suppose that M is an orthogonal, 2 by 2 matrix that has real number entries. Does M have to be a rotation matrix? If you believe that M must be a rotation matrix, explain why. If you do not believe that M has to he a rotation matrix, give an example. S. (16 points total) In this probiem you will ﬁt a quadratic polynomial to the data points: (0, 27), (1. 0), (2, 0). (3, 0) using the technique of Least Squares. (a) (4 points) A quadratic polynomial is a function of the form: ﬁx) = (2'12: + 17-): + 1:. Use the data points given above to set up a system of linear equations. Brieﬂy explain how you can tell that the system of equations is inconsistent. (b) (6 points) The least-squares solution of a system A} = E is the solution of the corresponding normal equation: ATM = ATE . Set up the normal equation for the system of linear equations that you formulated in Part (a). (c) (6 points} Find the equation of the quadratic polynomial that best ﬁts the data points given above. 6. (19 points total) In this problem. the matrix A will always be the matrix: —1 20 —34 “16 —_23 E in 409 2 4 4 l A‘ .7 7—5 _—=23 :9; 2 2 2 —_3 a i it 2 i 4 A It is possible to show that there is an invertible 4 by 4 matrix S such that: 1000 0—7 00 00 S'IAS = owo 0 0 . 3 Please note that it is not necesmy for you to calculate the matrix S, although if you believe that this will help you solve the problem you are welcome to do so. (a) (b) (d) (6 points) What are the eigenvalues of the matrix A? (6 points} Complete the table given below. (Note: The table has deliberawa been constructed with more spaces that you actually need, so don’t worry if you have space left over.) Eigenvalue of A Algebraic Multiplicity Geometric Multiplicity (4 points) Show that: det(A) = det(S‘ 1AS). {3 points) What is the determinant of A? Is A an invertible matrix? Explain how you know. 7. (10 points total) LetA be a symmetric r: by in matrix. That is, A = AT. Suppose that Al and 112 are distinct eigenvalues of A. That is. it, ¢ 11;. Suppose that i is an eigenveclor of A with eigenvalue 1:. Suppose that 17 is an eigenveetor of A with eigenvalue 3:. Show that the vectors 1? and i5 are orthogonal. Hint: Recall that fin? = 17717. ...
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