18.06 Spring 2006 - Review ProblemsThe following problems are intended to help you review for the final exam. Theyare not necessarily indicative of the types of questions that will appear on the exam.1. Suppose the following information is known about a matrix A:i) A200=24−6ii) A0−10=−2−46iii) A is symmetric.a) Is the nullspace of A zero?b) Is A invertible?c) Does A have linearly independent eigenvectors?d) Give a specific examples of a matrix A satisfying the above three properties andwhose eigenvalues add up to zero.2. Find all solutions to the linear system:w + 2x + y − 2z = 53w + 6x + 2y − z = 14−2w − 4x − y − z = −93. If A3= 2A2− A, w hat are the possible eigenvalues of A?14. Find the inverse of A =1 −1 02 1 30 2 1.5. a) Is Txyz=2x + y + 3y2− 4za linear transformation from R3to R2?Justify your answer.b) Let T : R2→ R3be a linear transformation such thatT10=12−1and T−11=253i) Calculate T3−1.ii) Is11−2in the range of T ?6. a) Give a 3 × 3 matrix A (not a diagonal matrix) with the following properties:i) AT= A−1.ii) det(A) = 1.b) Give a 3 × 3 matrix A (not a diagonal matrix) with the following prop e rties:i) AT= A.2ii) A2= A.iii) rk(A) = 1.c) Suppose A is 5 × 3 with orthonormal columns. Evaluate the following determi-nants:i) det(ATA).ii) det(AAT).iii) det(A(ATA)−1AT).d) For which value(s) of α ∈ R is A invertible?A =α 2 3−α α 03 2 5.7. Let A be real n × n matrix, A2= −In.a) Show A is invertible.b) Show n is even.c) Show A has no real eigenvalues.8. A =3 2 22 3 −2a) Find the SVD of A.b) Find orthonormal bases for the four fundamental subspaces of A.39. The matrixA =198 2 −22 5 4−2 4 5is the matrix of the orthogonal projection onto some subspace V ⊆ R3.a) Find an orthonormal basis for V .b) Find an orthonormal basis for V⊥.c) Find the matrix P of the orthogonal projection onto V⊥.10. Find the derivative of the function f(x) wheref(x) = det1 1 2 3 49 0 2 3 49 0 0 3 4x 2 0 0 49 0 0 0 4.11. In any year, 92% of deer in the forest remain there, while 8% find their way intothe suburbs (and people’s backyards, where they eat their shrubbery). In addition,88% of the deer in the suburbs remain there, while 12% are caught and returned tothe forest. Currently, 90% of the deer population is in the forest and the other 10%is in the suburbs. In the long run (after 20-50 years), what percentage of the deerpopulation can we expect is in the forest and what percentage is in the suburbs?(The total deer population is assumed to be constant throughout.)412. Let A =1 22 4.a) Find the eigenvalues of A.b) Give a factorization of A = QDQTwhere Q has orthonormal columns and D isdiagonal.c) As t → ∞, what is the limit of u(t) fordudt= −Au.given the initial condition u(0) =31.13. a) If possible, find an invertible matrix M such thatM−11 1 11 1 11 1 1M =1 1 11 2 21 2 2.If not possible, state why M cannot exist.b) For what values of c (if any) isA =−1 c 2c −4 −32 −3 4a symmetric positive definite matrix?514. Find the quadratic polynomial p(t) = a + bt + ct2that best fits the points(−2, −4), (−1, −1), (0, 0), (1, 0), (2, 0).15. Pnis the set of all polynomials of degree ≤ n. Let T : P2→ P1be the lineartransformationT (p(t)) = p0(t).Find the matrix A representing T with respect to the bases {t2+3t+1, t2−2, 3t+1}for P2and {t + 1, t + 2} for P1.16. Let A =1 −1 −11 1 01 0 1.a) Find the LU decomposition of A, if possible.b) Find the LDU decomposition of A, if possible.c) Find the QR decomposition of A, if
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