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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Transcribed Image Text:**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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