1. Let P3 be the space of polynomials of degree at most 3, i.e., P3 = {po+ p1x + p2x² + p3x% : po, P1, P2, P3 € R}. Let T : P3 → P3 be the mapping defined by Tf (x) = f"(x) – 4f'(x) +f(x). Show A = {1, 1+x, (1+x)², (1+x)³} is a basis for P3. Find the matrix representation of T with respect to the ordered basis A = (1, 1+x, (1+x)², (1+x)³). %3D %3D

1. Let P3 be the space of polynomials of degree at most 3, i.e., P3 = {po+ p1x + p2x² + p3x% : po, P1, P2, P3 € R}. Let T : P3 → P3 be the mapping defined by Tf (x) = f"(x) – 4f'(x) +f(x). Show A = {1, 1+x, (1+x)², (1+x)³} is a basis for P3. Find the matrix representation of T with respect to the ordered basis A = (1, 1+x, (1+x)², (1+x)³). %3D %3D

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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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

1. Let P3 be the space of polynomials of degree at most 3, i.e.,
P3 = {po+ P1x +p2x² + P3x* : po, P1, P2, P3 E R}.
Let T: P3 → P3 be the mapping defined by Tf (x) = f"(x) – 4f'(x) +f(x). Show
A = {1, 1+x, (1+x)², (1+x)³} is a basis for P3. Find the matrix representation
of T with respect to the ordered basis A = (1, 1+x, (1+x)², (1+x)³).
-

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