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Let V be the set of polynomials with real coefficients having degree at most 2. Prove that V is a vector space over the field of real numbers. Prove that the set {1,x,x²} is a basis of V and give matrix representations D1, D2:V → V of the first and second derivative mappings. Does D2 = D, • D,?

Let V be the set of polynomials with real coefficients having degree at most 2. Prove that V is a vector space over the field of real numbers. Prove that the set {1,x,x²} is a basis of V and give matrix representations D1, D2:V → V of the first and second derivative mappings. Does D2 = D, • D,?

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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**Polynomial Vector Space** Let \( V \) be the set of polynomials with real coefficients having degree at most 2. Prove that \( V \) is a vector space over the field of real numbers. Prove that the set \(\{1, x, x^2\} \) is a basis of \( V \) and give matrix representations \( D_1, D_2: V \to V \) of the first and second derivative mappings. Does \( D_2 = D_1 \circ D_1 \)?
Transcribed Image Text:**Polynomial Vector Space** Let \( V \) be the set of polynomials with real coefficients having degree at most 2. Prove that \( V \) is a vector space over the field of real numbers. Prove that the set \(\{1, x, x^2\} \) is a basis of \( V \) and give matrix representations \( D_1, D_2: V \to V \) of the first and second derivative mappings. Does \( D_2 = D_1 \circ D_1 \)?
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