**Polynomial and Subspaces**A polynomial is called **monic** if the coefficient of the highest degree term is 1. For example:- $ f(x) = x^3 + 2x + 3 $ and $ g(x) = x + 5 $ are monic polynomials.- $ h(x) = 5x^2 + x + 1 $ is not monic.Let $ W $ be the set of all monic polynomials in $ P_2(x) $.**Problem Statement:** Prove or disprove: Is $ W $ a subspace of $ P_2(x) $?In this problem, you need to determine if the set $ W $, which consists of monic polynomials with a degree of 2 or less, forms a subspace within the polynomial space $ P_2(x) $. A subspace must satisfy specific criteria, such as closure under addition and scalar multiplication, and contain the zero vector. Use these properties to either prove or disprove the subspace status of $ W $.

3. A polynomial is called monic if the coefficient of the highest degree term is 1. (Fór example, Ax) = x³ + 2x + 3 and g(x) =x+5 are monic but h(x) = 5x² + x + 1 is not monic). Let W be the se monic polynomials in P2(x). Prove or disprove: Is Wa subspace of P2(x)? %3D

3. A polynomial is called monic if the coefficient of the highest degree term is 1. (Fór example, Ax) = x³ + 2x + 3 and g(x) =x+5 are monic but h(x) = 5x² + x + 1 is not monic). Let W be the se monic polynomials in P2(x). Prove or disprove: Is Wa subspace of P2(x)? %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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Question

**Polynomial and Subspaces**

A polynomial is called **monic** if the coefficient of the highest degree term is 1. For example:
- $ f(x) = x^3 + 2x + 3 $ and $ g(x) = x + 5 $ are monic polynomials.
- $ h(x) = 5x^2 + x + 1 $ is not monic.

Let $ W $ be the set of all monic polynomials in $ P_2(x) $.

**Problem Statement:** Prove or disprove: Is $ W $ a subspace of $ P_2(x) $?

In this problem, you need to determine if the set $ W $, which consists of monic polynomials with a degree of 2 or less, forms a subspace within the polynomial space $ P_2(x) $. A subspace must satisfy specific criteria, such as closure under addition and scalar multiplication, and contain the zero vector. Use these properties to either prove or disprove the subspace status of $ W $.

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