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etermine whether the set S = {1-3xr?, 2x+5x²,1-x+3x²} is basis for P,(R).

etermine whether the set S = {1-3xr?, 2x+5x²,1-x+3x²} is basis for P,(R).

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement:** Determine whether the set \( S = \{1 - 3x^2, 2x + 5x^2, 1 - x + 3x^2\} \) is a basis for \( P_2(\mathbb{R}) \). **Explanation:** In the above problem, the set \( S \) consists of three polynomials: \( 1 - 3x^2 \), \( 2x + 5x^2 \), and \( 1 - x + 3x^2 \). The task is to determine whether this set forms a basis for the vector space \( P_2(\mathbb{R}) \), which is the space of all polynomials of degree less than or equal to 2 with real coefficients. To be a basis, the set must satisfy two conditions: 1. It must be linearly independent. 2. It must span the vector space \( P_2(\mathbb{R}) \).
Transcribed Image Text:**Problem Statement:** Determine whether the set \( S = \{1 - 3x^2, 2x + 5x^2, 1 - x + 3x^2\} \) is a basis for \( P_2(\mathbb{R}) \). **Explanation:** In the above problem, the set \( S \) consists of three polynomials: \( 1 - 3x^2 \), \( 2x + 5x^2 \), and \( 1 - x + 3x^2 \). The task is to determine whether this set forms a basis for the vector space \( P_2(\mathbb{R}) \), which is the space of all polynomials of degree less than or equal to 2 with real coefficients. To be a basis, the set must satisfy two conditions: 1. It must be linearly independent. 2. It must span the vector space \( P_2(\mathbb{R}) \).
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