Let P3(R) be the space of polynomials in x with real coefficients, having degree atmost 3. Let T : P3(R) → P3(R) be the linear transformation defined byT(a + bx + cx? + dx*) :(2а — b + 3с— d) + (2а — b + Зс — d)х + (2а — b - Зс + d)л? + (2а — b — Зс + d)д3.What are the kernel and range of T?Select one:Ker(T) = span{1 – 2.x, r2 – 3r};Rng(T) = span{1+x,x² + x³}.-O a.Ker(T) = span{1+2x, x² + 3.x³};Rng(T) = span{1+x,x² + x³}.O b.Ker(T) = span{1 – 2.x, x² + 3.x³};Rng(T) = span{1+x,x² – x³3}.Ker(T) = span{1+2x, a² – 3.r3};Rng(T) = span{1 – x, x2 + x3}.Od.

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Let P3(R) be the space of polynomials in r with real coefficients, having degree at most 3. Let T : P3(R) → P3(R) be the linear transformation defined by T(a + bx + cx? + dx) = (2а — b + Зс — d) + (2а — b + Зс — d)я + (2а — b — Зс+ d)2? + (2а —b - Зс + d)a*. What are the kernel and range of T? Select one: Ker(T) = span{1 – 2x, 22 – 3x}; Rng(T) = span{1+x,x² + x³}. | O a. span{1+2x, x2 + 3x³}; Ker(T) = span{1 – 2.x, x² + 3.r³}; Ker(T) = span{1+2.x, x² – 3x³}; Rng(T) = span{1+x,x² + x*}. Rng(T) = span{1+x,x² – x³}. Ker(T) Ob. | Ос. Rng(T) = span{1 – x, r² + x³}. d.

Let P3(R) be the space of polynomials in r with real coefficients, having degree at most 3. Let T : P3(R) → P3(R) be the linear transformation defined by T(a + bx + cx? + dx) = (2а — b + Зс — d) + (2а — b + Зс — d)я + (2а — b — Зс+ d)2? + (2а —b - Зс + d)a. What are the kernel and range of T? Select one: Ker(T) = span{1 – 2x, 22 – 3x}; Rng(T) = span{1+x,x² + x³}. | O a. span{1+2x, x2 + 3x³}; Ker(T) = span{1 – 2.x, x² + 3.r³}; Ker(T) = span{1+2.x, x² – 3x³}; Rng(T) = span{1+x,x² + x}. Rng(T) = span{1+x,x² – x³}. Ker(T) Ob. | Ос. Rng(T) = span{1 – x, r² + x³}. d.

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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Question

Let P3(R) be the space of polynomials in x with real coefficients, having degree at
most 3. Let T : P3(R) → P3(R) be the linear transformation defined by
T(a + bx + cx? + dx*) :
(2а — b + 3с— d) + (2а — b + Зс — d)х + (2а — b - Зс + d)л? + (2а — b — Зс + d)д3.
What are the kernel and range of T?
Select one:
Ker(T) = span{1 – 2.x, r2 – 3r};
Rng(T) = span{1+x,x² + x³}.
-
O a.
Ker(T) = span{1+2x, x² + 3.x³};
Rng(T) = span{1+x,x² + x³}.
O b.
Ker(T) = span{1 – 2.x, x² + 3.x³};
Rng(T) = span{1+x,x² – x³3}.
Ker(T) = span{1+2x, a² – 3.r3};
Rng(T) = span{1 – x, x2 + x3}.
Od.

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