5. Let P₂(C) be the vector space of polynomials of degree less than or equalto 2 over C. Let C³ be the vector space over C. By using linear extensionmethod, show that P₂(C) = C³.6. Let P₁(C) be the vector space of polynomials of degree less than or equalto 1 over R and C³ be the vector space over R. Let a = {1,i, x, zi} bethe basis for P₂ (C) and 3 = {(1, 1, 1), (2, 3,0), (i, 1,0), (0, 0, i)} be arbitraryvectors in C³. Determine whether linear transformation T: P₁(C) → C³exists by using linear extension method.

5) Let P2ℂ be the vector space of all polynomials of degree less than or equal to 2 over ℂ. and ℂ3…

Answered: 5. Let P2(C) be the vector space of…

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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4) Let V be the vector space of polynomials of degree <2 and let p(x) = ax² + bx + c. Find the adjoint operator (T*@)(p) for T:V → V and o:V → R given by d) Tp(x) = x²p(x) + x³p'(x), @(p) = p"(1).
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"On R², define the operations of addition and scalar multiplication as follows: (X₁, X₂) + (y₁ Y₂) = (x₂ + y₂ +1, x₂ + y₂ + 1), c(x₁, x₂) = (Cx₂ + C-1, ₂+c-1). Then R² with these operations forms a vector space." Note that here the zero vector is (-1,-1) and the additive inverse of (x1, x2) is (-X₁-2, -x₂-2), not (-x₂-x₂)- a) Show that (-1, -1) acts as the zero vector under these definitions.
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5. Let P2(C) be the vector space of polynomials of degree less than or equal to 2 over C. Let C3 be the vector space over C. By using linear extension method, show that P₂(C) = C³.