Let P, denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: P3 → P, be the function that sends a polynomial to its derivative. That is, D(p(x)) = p' (x) for all polynomials p(x) E P3. Is Da linear transformation? Let p(x) = azx3+ a2x² + a1x + ao and q(x) = bzx³ + b2x² + b,x + bo be any two polynomials in P3 and ce R. a. D(p(x) + q(x)) = (Enter az as a3, etc.) D(p(x)) + D(q(x)) Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) E P3? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c e R and all p(x) E P3? choose C. Is Da linear transformation? choose

Let P, denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: P3 → P, be the function that sends a polynomial to its derivative. That is, D(p(x)) = p' (x) for all polynomials p(x) E P3. Is Da linear transformation? Let p(x) = azx3+ a2x² + a1x + ao and q(x) = bzx³ + b2x² + b,x + bo be any two polynomials in P3 and ce R. a. D(p(x) + q(x)) = (Enter az as a3, etc.) D(p(x)) + D(q(x)) Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) E P3? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c e R and all p(x) E P3? choose C. Is Da linear transformation? choose

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.6: Introduction To Linear Transformations

Problem 38EQ

See similar textbooks

Similar questions

In Exercises 3-6, prove that the given transformation is a linear transformation, using the definition (or the Remark following Example 3.55). 6.
In Exercises 3-6, prove that the given transformation is a linear transformation, using the definition (or the Remark following Example 3.55). T[xy]=[yx+2y3x4y]
In Exercises 7-10, give a counterexample to show that the given transformation is not a linear transformation. 7.
The figure shows three outline versions of the letter F. The second one is obtained from the first by shrinking horizontally by a factor of 0.75, and the third is obtained from the first by shearing horizontally by a factor of 0.25. a Find a data matrix D for the first letter F. b Find a transformation matrix T that transforms the first F into the second. Calculate TD, and verify that this is a data matrix for the second F. c Find the transformation matrix S that transforms the first F into the third. Calculate SD, and verify that this is a data matrix for the third F.
In Exercises 7-10, give a counterexample to show that the given transformation is not a linear transformation. 9.
In Exercises 7-10, give a counterexample to show that the given transformation is not a linear transformation. 10.
In Exercises 7-10, give a counterexample to show that the given transformation is not a linear transformation. 8.
For the linear transformation from Exercise 38, find a T(0,1,0,1,0), and b the preimage of (0,0,0), c the preimage of (1,1,2). Linear Transformation Given by a Matrix In Exercises 33-38, define the linear transformation T:RnRmby T(v)=Av. Find the dimensions of Rnand Rm. A=[020201010112221]
In Exercises 11-14, find the standard matrix of the linear transformation in the given exercise. T[xyz]=[x+zy+zx+y]
In Exercises 11-14, find the standard matrix of the linear transformation in the given exercise. 11.
For the linear transformation from Exercise 37, find a T(1,0,2,3), and b the preimage of (0,0,0). Linear Transformation Given by a Matrix In Exercises 33-38, define the linear transformations T:RnRm by T(v)=Av. Find the dimensions of Rn and Rm. A=[012114500131]
In Exercises 11-14, find the standard matrix of the linear transformation in the given exercise. T[xyz]=[xy+z2x+y3z]