Let F be a field and U = Fx] be the vector space of polynomials in...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Let F be a field and U = Fx] be the vector space of polynomials in one variable with coefficients in F. We define a linear operator D:U U by sending a polynomial to its derivative. More precisely, D is defined by the properties D(1) = 0, D(x) = 1, as well as the Leibniz rule, which makes use of the product of polynomials: D(fg) = (Df)g + fDg, Vf,g E F[x]. 1. Determine all eigenvalues and eigenvectors of D, and conclude that there is no nonzero element f e U satisfying the differential equation Df = f. Much in same way that we adjoin square roots of -1 to the real numbers in order to construct the complex numbers, we may adjoin a nontrivial solution to Df = f: we add an extra variable or "indeterminate" e, enlarging our vector space from U to V = F\x, e], the polynomials in two variables. We then extend our operator D to the larger vector space V by requiring the same conditions above (extending the Leibniz rule to all of Fx, e) as well as the new condition D(e) = e. We "abuse notation" by using the same name, D, for the original operator on U as well as its extension to V. 2. Determine all eigenvalues, eigenvectors, and generalized eigenvectors of the operator D on V. 3. Let Sc V be the subspace of solutions to the homogeneous differential equation f" – 4f" + 5f – 2f = 0, in other words, S = Null(D3 – 4D2 + 5D - 2). Find a basis for S putting D into Jordan form. Let F be a field and U = Fx] be the vector space of polynomials in one variable with coefficients in F. We define a linear operator D:U U by sending a polynomial to its derivative. More precisely, D is defined by the properties D(1) = 0, D(x) = 1, as well as the Leibniz rule, which makes use of the product of polynomials: D(fg) = (Df)g + fDg, Vf,g E F[x]. 1. Determine all eigenvalues and eigenvectors of D, and conclude that there is no nonzero element f e U satisfying the differential equation Df = f. Much in same way that we adjoin square roots of -1 to the real numbers in order to construct the complex numbers, we may adjoin a nontrivial solution to Df = f: we add an extra variable or "indeterminate" e, enlarging our vector space from U to V = F\x, e], the polynomials in two variables. We then extend our operator D to the larger vector space V by requiring the same conditions above (extending the Leibniz rule to all of Fx, e) as well as the new condition D(e) = e. We "abuse notation" by using the same name, D, for the original operator on U as well as its extension to V. 2. Determine all eigenvalues, eigenvectors, and generalized eigenvectors of the operator D on V. 3. Let Sc V be the subspace of solutions to the homogeneous differential equation f" – 4f" + 5f – 2f = 0, in other words, S = Null(D3 – 4D2 + 5D - 2). Find a basis for S putting D into Jordan form.
Expert Answer:
Related Book For
Probability and Statistics
ISBN: 978-0321500465
4th edition
Authors: Morris H. DeGroot, Mark J. Schervish
Posted Date:
Students also viewed these mathematics questions
-
Let V = P(R) be the vector space of polynomials with coefficients in R and let X be a finite subset of V. Prove that there exists an element of V which cannot be expressed as a linear combination of...
-
Let f be a twice differentiable functional on some open interval S containing x0. For every x1 S, there exists some between x0 and x1 such that f (x1) = f(xo) +f"lxo]x+f"x?
-
Let f be a (n + 1)-times differentiable functional on some open interval S containing x0. For every x S - x0, there exists some between x0 and x0 + x such that +!")[xo)x" +7n+1)!j"h+1)|N].xn+1 +
-
Mr. Shroff can use 360 feedback for all of the following purposes EXCEPT: Multiple Choice Job redesign Training program objectives Feedback and performance improvement Employee development Employee...
-
A supplier of parts to an assembly plant in the household appliance industry is required to make deliveries on a just-in-time basis (daily). For one of the parts that must be delivered, the daily...
-
Mach IV Audio uses a periodic inventory system. One of the stores most popular products is an MP3 car stereo system. The inventory quantities, purchases, and sales of this product for the most recent...
-
Randomly permute each row of the matrix below using R. \[\left[\begin{array}{llll}1 & 1 & 1 & 1 \\2 & 2 & 2 & 2 \\3 & 3 & 3 & 3 \\4 & 4 & 4 & 4\end{array} ight]\] You can construct the matrix below...
-
Royston Inc. is a large food processing company. It processes 120,000 kilograms of peanuts in the Peanuts Department at a cost of $160,000 to yield 10,000 kilograms of product A, 60,000 kilograms of...
-
Cherokee Incorporated is a merchandiser that provided the following information: Number of units sold Selling price per unit Variable selling expense per unit Variable administrative expense per unit...
-
Gibson Agency Case: 1. Calculate and present the budgeted profit for each of Gibson's clients for each of the years 2016 through 2019, using the current costing system (i.e., the one described in the...
-
Describe how psychographic applications in marketing combine values, personality, and lifestyle variables. Discuss potential limitations of psychographic data in a marketing context.
-
Have a mock face-to-face discussion with a friend, coworker, or family member about a descriptive subject of your choice. Afterward, try to replicate the same conversation via text or email. What...
-
Compute the impact on the money multiplier of a fall in the currency-to-deposit ratio from 10 percent to 8 percent when the reserve requirement is 10 percent of deposits, and banks desired excess...
-
Problems 35 through 40 show a free-body diagram. For each: a. Identify the direction of the acceleration vector au and show it as a vector next to your diagram. Or, if appropriate, write a = 0. b. If...
-
Comparative Analysis Case adidas and Puma The financial statements of adidas (DEU) and Puma (DEU) are presented in Appendices B and C, respectively. The complete annual reports, including the notes...
-
Consider the problem of carbon dioxide emissions. We will abstract away from the problem slightly, assuming there are polluters and consumers in two regions, the \(\mathrm{OECD}(\mathrm{O})\) and the...
-
Calculate complexity of the function and return value in-term of N as an argument, for fun3, fun2 and fun1, with all steps and details int fun3(int & N) { N--; for (i = 1; i
-
A 6-lb shell moving with a velocity ?? v0k explodes at point D into three fragments which hit the vertical wall at the points indicated. Fragments A, B, and C hit the wall 0.010 s, 0.018 s, and 0.012...
-
Suppose that two players A and B are trying to throw a basketball through a hoop. The probability that player A will succeed on any given throw is p, and he throws until he has succeeded r times. The...
-
Suppose that X1, . . . ,Xn are i.i.d. random variables, each of which has a continuous distribution with median m. Let Yn = max{X1, . . . , Xn}. Determine the value of Pr(Yn > m).
-
Describe how to convert a random sample U1, . . . , Un from the uniform distribution on the interval [0, 1] to a random sample of size n from the uniform distribution on the interval [a, b].
-
Carry out an Internet search to find more examples of project success and failure. From your search, are there any common themes in each? What are the implications of success and failure in each case?
-
How successful are government contracting arrangements? How do these compare, for instance, with the arrangements BAA had with their suppliers in the construction of T5 at Heathrow (Project...
-
What is the role of brainstorming and how might it be used to greatest effect?
Study smarter with the SolutionInn App