Р(1)p(2)p(3)mation, and find ker (T). Recall that P3 is the space of all polynomials of degree at most8. Let T : P3 → R³ be defined by T (p(x))Show that T is a linear transfor-3, P3 := {p(x) = ao + a1x + a2² + azx*| a1, a2, a3, a4 E R}.

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Answered: p(1) Р(2) p(3) mation, and find ker…

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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This is a question from a linear algebra course: Let V = R[X]3, the polynomials of degree at most three, and B = {1, X, X2, X3}. Show what the image under fB is of:• the four basic elements: P1(X) = 1, P2(X) = X, P3(X) = X2 and P4(X) = X3• P(X) = 2 + 6X + 3X2 + 4X3
f: X→ Y and g: Y→V be bijective functions. Show the following: ) gof: X → V is injective. ) gof: X→ V is surjective. ) (gof)-¹ = f¹og-¹ v injective if x1 #xn → v(x1) = v(xn)
1. Determine whether the following polynomials span P. =1+x, p, = 1 – x, P2 P3D1+x+x², P, = 2-x² 3D2 3:28 PM 豆 IA 2/23/2022 e Type here to search WQHD | 165HZ NOG ZEPHYRUS 100% DC-P3 PANTONE Validated F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 ESC E3 O/京 DELETE AURA 23 24 4 @ % & 7 { } TAB Q W E T Y P. [ 1O 15 林
Assume the polynomial P5(x) interpolates a function f(x) at the six data points (xi, f(xi)) with x-coordinates ₁ = 0.0, 2 0.0, x2 = 0.2, x3 = 0.4, x4 = 0.6, x5 = 0.8, x6 = 1.0. Assume that the interpolation error at x = 0.3 is |f(0.3) - P5(0.3)| = 0.01. Estimate the new interpolation error |ƒ(0.3) — P7(0.3)| that would result if two additional interpolation points - (x7, y7) = (0.1, f(0.1)) and (x8, ys) = (0.5, f(0.5)) are added. What assumptions have you made to produce this estimate?

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p(1) Р(2) p(3) mation, and find ker (T). Recall that P3 is the space of all polynomials of degree at most Let T : P3 → R³ be defined by T (p(x)) = Show that T is a linear transfor- 3, P3 := {p(x) = ao + a1x + a2x² + azx*| a1, a2, a3, a4 € R}.