Practice Final 11. Answer the following questions “true,” “false,” or“I don’t know.” Five points will b e given for thecorrect answer, and three points will be given for“I don’t know.”(a) The functions f(x) = x and g(x) = |x| arelinearly independent on the interval [−1, 1].(b) Let A be a matrix of rank 2 and B bea matrix of rank 1, both n × n. Thenrk(AB) = 1.(c) The function f(x) = ln |x| is piecewise con-tinuous on R.(d) Let P be an orthogonal projection matrixfor a subspace V of R3. Then P is diago-nalizable.(e) For any func tions p(x) and q(x) which arecontinuous on R, there exists a functiony(x) such that y00+ py0+ qy = 0, y(0) = 0,and y0(1) = 1.2. Consider the following system of differentialequations.x0(t) =13−1 −2 −7−4 1 20 0 3x(t) (1)(a) Find the general solution to (1).(b) Find the solution to (1) such thatx(0) =−110.3. Consider the partial differential equationuxx+ uxt+ ut= 0.(a) Transform this equation into a collection ofordinary differential equations using sepa-ration of variables.(b) Find all solutions to your ordinary dif-ferential equations from above such thatu(0, t) = u(π, t) = 0.4. Consider the following differential equation.y000− 3y0− 2y = 0 (2)(a) Write (2) as a system of first-order differen-tial equations.(b) Give the general solution to (2).(c) Give the solution to (2) such that y(4) = 2,y0(4) = 0, and y00(4) = −1.5. Consider the following linear transformation T :P2→ P2.T (a0+a1x+a2x2) = a0+a2−a1x+(a0+a1+a2)x2(a) Give a basis for the kernel of T .(b) Give the matrix for T relative to the basis{1, x − 1, (x − 1)2}.Practice Final 21. Answer the following questions “true,” “false,” or“I don’t know.” Five points will b e given for th ecorrect answer, and three points will be given for“I don’t know.”(a) The functionf(x) =−1 x < 08 x = 01 x > 0is piecewise continuous on R.(b) Let A be an n × n matrix whose only eigen-value in C is 5. Then rk(A) > rk(A − 5I).(c) The functions f (x) = excos x and g(x) =exsin x are linearly dependent on the inter-val [−2π, 2π].(d) Let A and B be two 2 × 2 matrices. Then(AB)T= ATBT.(e) Let p(x) and q(x) be continuous on R. Thenthere exists a unique solution to the differ-ential equation p(x)y00+ 3y0− q(x)y = 0 onthe interval (−1, 1) su ch that y(0) = 1 andy0(0) = −3.2. Consider the following system of differentialequations.x0(t) =2 1 0−2 0 01 2 −1x(t) (3)(a) Find the general solution to (3).(b) Give the solution to (3) satisfying the initialconditionx(0) =10−1.13. Transform the partial differential equationuxt− αutt= 0into a collection of ordinary differential equationsusing separation of variables.4. Solve the following heat conduction problem in-volving a wire of length 10.9uxx= utu(0, t) = 10u(10, t) = 20u(x, 0) = sinπx55. For any x ∈ [−π, π), letf(x) = x − cos x.For x /∈ [−π, π), extend f so that it is p eriodicof period 2π. Give the Fourier series for f .6. Define an inner product on P2byhp, qi = p(0)q(0) + p(1)q(1) + p(2)q(2) (4)for all p, q, ∈ P2.(a) Give an orthogonal basis for the subspaceS = {p ∈ P2| p(1) = 0}with respect to the inner product (4).(b) Let T : P2→ P2be orthogonal projectiononto S with respect to the inner product(4). Give the matrix for T relative to theordered basis {1, x − 1, x2−
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