Show that A is an automorphism and find its inverse.Problem 2Let P3[x] be the vector space of all polynomial of order at most3. Suppose that B = {1, (x + 1), (x + 1)², (x + 1)³}. Show that B is a basis for P3|z]. Givenp(x) = x³ + x, find [p(x)]B.

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Answered: Show that A is an automorphism and find…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let T be the linear transformation from the space of degree three or less polynomials to itself defined by the following formula: T(ax³ + bx² + cx + d) = (a − b)x³ + (a - b)x² + (2c+ d)x + (2c+ d) T. Find a basis for the image of
Let T be the linear transformation from M2,2 to the space of polynomials defined by: • Evaluate: b T = (a+b)x+c d T T 5 1 3 1 1 Based off these calculations, which of and 0 5 are in the null space of -1 3 T? • Name at least two specific polynomials which are in the image of T.
Let W = {(x1, x2, x3) € R³ : x₁ + 2x2 3x3 = 0} (a) Find a basis of W. (b) Give an example of a subspace L in R³ such that R³ = WL. Justify your statement.
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Suppose T: P2→R is a linear transformation whose action on a basis for P2 is as follows: 6 -8 T(x2+x+1) = -1 T(x²+x+2)=|-6 X 2 4 -4 | Determine whether T is one-to-one and/or onto. If it is not one-to-one, show this by providing two polynomials that have the same image under T. If T is not onto, show this by providing a vector in R that is not in the image of T. Use the A' character to indicate an exponent and x as the variable for polynomials, e.g. 5x^2-2x+1 T is not one-to-one: T(0)= 0 and T(0)=|0 T is onto
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.

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Show that A is an automorphism and find its inverse. Problem 2 Let P3[x] be the vector space of all polynomial of order at most 3. Suppose that B = {1, (x + 1), (x + 1)², (x + 1)³). Show that B is a basis for P3|z]. Given p(x)= x³ + x, find [p(x)]B.