6. Let space V be the space of all polynomials of degree 3: V = {f(x) = ax + ba² + cx + d}. %3D Space W is defined as all those polynomials f from V such that f(0) = 0: W = {f €V: f(0) = 0} Question a. Prove that space W is a linear subspace of V. Question b. Let D(f(x)) be a transformation of V defined as follows: df D(f(x)) = 2f(x) – 3 (x) dx Verify whether D is a linear transformation. If yes then find the kernel of D.
6. Let space V be the space of all polynomials of degree 3: V = {f(x) = ax + ba² + cx + d}. %3D Space W is defined as all those polynomials f from V such that f(0) = 0: W = {f €V: f(0) = 0} Question a. Prove that space W is a linear subspace of V. Question b. Let D(f(x)) be a transformation of V defined as follows: df D(f(x)) = 2f(x) – 3 (x) dx Verify whether D is a linear transformation. If yes then find the kernel of 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
6.
Transcribed Image Text:6. Let space V be the space of all polynomials of degree 3:
V = {f(x) = ar + bx2 + cx + d}.
Space W is defined as all those polynomials f from V such that f(0) = 0:
W = {f € V: f(0) = 0}
Question a. Prove that space W is a linear subspace of V.
Question b. Let D(f(x)) be a transformation of V defined as follows:
df
D(f(x)) = 2f(x) – 3 (x)
d.x
Verify whether D is a linear transformation. If yes then find the kernel
of D.
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