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Consider the following polynomials in P • P4(x)=x+1; Ps(x)=²+1; • Pu(x)=2²³ +1. P₁(x) = x³ + x² +1; • P₂(x)=2²³+x; P₁(x)=x²+x; (a) Find a basis B of Span(P₁(x),..., Pa(z)). (b) Determine whether there exists a linear transformation f: P, → P that satisfies all the following: • f(pi(x)) = 2² +1; • 1 (P₂(x))=x²²; • f(ps(x)) = 2² +2²; • f(pa(z))=2¹; • f(ps(x)) = a¹ +²+2; • f(ps(x)) = 0.

Consider the following polynomials in P • P4(x)=x+1; Ps(x)=²+1; • Pu(x)=2²³ +1. P₁(x) = x³ + x² +1; • P₂(x)=2²³+x; P₁(x)=x²+x; (a) Find a basis B of Span(P₁(x),..., Pa(z)). (b) Determine whether there exists a linear transformation f: P, → P that satisfies all the following: • f(pi(x)) = 2² +1; • 1 (P₂(x))=x²²; • f(ps(x)) = 2² +2²; • f(pa(z))=2¹; • f(ps(x)) = a¹ +²+2; • f(ps(x)) = 0.

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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Consider the following polynomials in P • P4(x)=x+1; Ps(x)=x²+1; • Pa(z)=2³ +1. • P₁(x) = x³ + x² +1; • P₂(x)= x³ + x; • P₁(x) = x² + x; (a) Find a basis B of Span (P₁(x),..., P(x)). (b) Determine whether there exists a linear transformation f : P², → PZ₂ that satisfies all the following: • f(p₁(x)) = x² + 1; • f(p₂(x)) = x²; • f(ps(x)) = x² + 2; • f(ps(x)) = x²; • f(ps(x)) = x² +2²+2; • f(ps(x)) = 0.
Transcribed Image Text:Consider the following polynomials in P • P4(x)=x+1; Ps(x)=x²+1; • Pa(z)=2³ +1. • P₁(x) = x³ + x² +1; • P₂(x)= x³ + x; • P₁(x) = x² + x; (a) Find a basis B of Span (P₁(x),..., P(x)). (b) Determine whether there exists a linear transformation f : P², → PZ₂ that satisfies all the following: • f(p₁(x)) = x² + 1; • f(p₂(x)) = x²; • f(ps(x)) = x² + 2; • f(ps(x)) = x²; • f(ps(x)) = x² +2²+2; • f(ps(x)) = 0.
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