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7. Prove that p, (t) = 1+t², p2(t) = t – 3t2, p3(t) = 1 +t – 3t, is a basis for P2.

7. Prove that p, (t) = 1+t², p2(t) = t – 3t2, p3(t) = 1 +t – 3t, is a basis for P2.

Advanced Engineering Mathematics
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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**Problem Statement:** Prove that the polynomials \( p_1(t) = 1 + t^2 \), \( p_2(t) = t - 3t^2 \), and \( p_3(t) = 1 + t - 3t^2 \) form a basis for the vector space \( P_2 \), the space of all polynomials of degree at most 2. **Explanation:** In this problem, we are tasked with showing that three specific polynomials form a basis for the vector space \( P_2 \). A set of polynomials forms a basis if they are linearly independent and span the space \( P_2 \). 1. **Linear Independence:** To prove linear independence, we need to show that if \( a(p_1(t)) + b(p_2(t)) + c(p_3(t)) = 0 \) (the zero polynomial), then \( a = b = c = 0 \). 2. **Spanning \( P_2 \):** To show these polynomials span \( P_2 \), we must demonstrate that any polynomial \( q(t) \) of degree at most 2 can be written as a linear combination of \( p_1(t) \), \( p_2(t) \), and \( p_3(t) \). This involves setting an arbitrary polynomial \( q(t) = x + yt + zt^2 \) and expressing it as \( q(t) = a(p_1(t)) + b(p_2(t)) + c(p_3(t)) \) by solving for coefficients \( a \), \( b \), and \( c \).
Transcribed Image Text:**Problem Statement:** Prove that the polynomials \( p_1(t) = 1 + t^2 \), \( p_2(t) = t - 3t^2 \), and \( p_3(t) = 1 + t - 3t^2 \) form a basis for the vector space \( P_2 \), the space of all polynomials of degree at most 2. **Explanation:** In this problem, we are tasked with showing that three specific polynomials form a basis for the vector space \( P_2 \). A set of polynomials forms a basis if they are linearly independent and span the space \( P_2 \). 1. **Linear Independence:** To prove linear independence, we need to show that if \( a(p_1(t)) + b(p_2(t)) + c(p_3(t)) = 0 \) (the zero polynomial), then \( a = b = c = 0 \). 2. **Spanning \( P_2 \):** To show these polynomials span \( P_2 \), we must demonstrate that any polynomial \( q(t) \) of degree at most 2 can be written as a linear combination of \( p_1(t) \), \( p_2(t) \), and \( p_3(t) \). This involves setting an arbitrary polynomial \( q(t) = x + yt + zt^2 \) and expressing it as \( q(t) = a(p_1(t)) + b(p_2(t)) + c(p_3(t)) \) by solving for coefficients \( a \), \( b \), and \( c \).
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