Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN: 9781305658004
Author: Ron Larson
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Related questions
Question
Transcribed Image Text:Let C (R) be the vector space of "smooth" functions, i.e., real-valued functions f(x) in the variable x that have infinitely many derivatives at all
points x Є R.
Let D C (R) → C∞ (R) and D² : C∞ (R) → C∞ (R) be the linear transformations defined by the first derivative D(f(x)) = f'(x) and the
second derivative D² (f(x)) = ƒ"(x).
a. Determine whether the smooth function g(x) = 8e-3 is an eigenvector of D. If so, give the associated eigenvalue. If not, enter NONE.
Eigenvalue =
b. Determine whether the smooth function h(x) = sin(5x) is an eigenvector of D2. If so, give the associated eigenvalue. If not, enter NONE.
Eigenvalue =
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by stepSolved in 2 steps
Knowledge Booster
Similar questions
- Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].arrow_forward, Let C" (R) be the vector space of "smooth" functions, i.e., real-valued functions f(x) in the variable z that have infinitely many derivatives at all points x E R. Let D: C* (IR) → C¤(R) and D² : C∞ (R) → C°(R) be the linear transformations defined by the first derivative D(f(x)) = f'(x) and the second derivative D²(f(x)) = f"(x). a. Determine whether the smooth function g(x) = 7e1z is an eigenvector of D. If so, give the associated eigenvalue. If not, enter NONE. Eigenvalue = b. Determine whether the smooth function h(x) = sin(9x) is an eigenvector of D2. If so, give the associated eigenvalue. If not, enter %3D NONE. Eigenvalue =arrow_forwardShow that t, e^t, and sin(t) are linearly independent.arrow_forward
- Let f(w)f(w)be a function of vector ww, i.e. f(w)=1/(1+e−wTx). Determine the first derivative and matrix of second derivatives of ffwith respect to w ?arrow_forwardLet denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)arrow_forwardSuppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the equation r1(t)=r2(t) may not produce all the points where the graphs of these functions intersect.arrow_forward
- Find the derivative of the vector function r(t) tax (btc), where a = (2, -5,-2), b = (4, 2,-5), and c = (3,-2,-2). r' (t) = (arrow_forwardSuppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the equation r1(t)=r2(t) may not produce all the points where the graphs of these functions intersect. Please Provide Unique Answer. Thank you!arrow_forwardLet r(t) be a vector-valued function such that the magnitude of r(t) does not change over time. Use derivatives to show that the derivative r'(t) is perpendicular to the function r(t) for all times t.arrow_forward
- Let L be the line in R2 through the points [:] } Find a linear functional f and a real number d such that L = [f:d].arrow_forwardf(x, 3) In(x2 + Yof function P(3, 4) at the point of = (9, 12) Directional derivative in the direction of the vector to you.arrow_forward(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of -Vf(x). Let 8 be the angle between Vf(x) and unit vector u. Then Duf = |Vf||---Select--- . Since the minimum value of ---Select--- is direction of u is ---Select--- ✓the direction of Vf (assuming Vf is not zero). (b) Use the result of part (a) to find the direction in which the function f(x, y) = x¹y x²y³ decreases fastest at the point (1, -1). occurring, for 0 ≤ 0 < 2π, when 8 = the minimum value of Duf is -|Vf, occurring when thearrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning