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6. Let space V be the space of all polynomials of degree 3: V = {f(x) = ax³ + ba² + 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) – 32 (2) dr 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) = ax³ + ba² + 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) – 32 (2) dr Verify whether D is a linear transformation. If yes then find the kernel of 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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6. Let space V be the space of all polynomials of degree 3: V = {f(x) = ax³ + ba² + cx + d}. Space W is defined as all those polynomials f from V such that f(0) = 0: %3D 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: D(f(x)) = 2f(x) – 3 (2) df dr Verify whether D is a linear transformation. If yes then find the kernel of D.
Transcribed Image Text:6. Let space V be the space of all polynomials of degree 3: V = {f(x) = ax³ + ba² + cx + d}. Space W is defined as all those polynomials f from V such that f(0) = 0: %3D 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: D(f(x)) = 2f(x) – 3 (2) df dr Verify whether D is a linear transformation. If yes then find the kernel of D.
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