3. Let V denote the vector space of all functions f: R→ R, equipped with addition +: V x V → V defined via (f+g)(x) = f(x) + g(x), x = R, and scalar multiplication: Rx V → V defined via (\. f)(x) = f(x), xER". Now let W= {f: RR: f(x) = ax + b for some a, b = R}, i.e. the space of all linear functions R→ R. (a) Show that W is a subspace of V. (You may assume that V is a vector space). [3] [4] (b) Find a basis for W. You should prove that it is indeed a basis.
3. Let V denote the vector space of all functions f: R→ R, equipped with addition +: V x V → V defined via (f+g)(x) = f(x) + g(x), x = R, and scalar multiplication: Rx V → V defined via (\. f)(x) = f(x), xER". Now let W= {f: RR: f(x) = ax + b for some a, b = R}, i.e. the space of all linear functions R→ R. (a) Show that W is a subspace of V. (You may assume that V is a vector space). [3] [4] (b) Find a basis for W. You should prove that it is indeed a basis.
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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Could you answer the question attached please?
![3. Let V denote the vector space of all functions ƒ : R → R, equipped with addition + : V × V → V defined
via (f+g)(x) = f(x) + g(x), x ≤ R, and scalar multiplication : R x V → V defined via (\ · ƒ)(x) = Xƒ(x),
xER".
Now let W =
{ƒ: R→R : f(x) = ax + b for some a, b ≤ R}, i.e. the space of all linear functions R → R.
(a) Show that W is a subspace of V. (You may assume that V is a vector space).
[3]
(b) Find a basis for W. You should prove that it is indeed a basis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8cb4a0f-b6a4-419b-8ce2-2ca5d96b779c%2F32d46714-b9da-4df6-ae04-914a100ce8f4%2Fjj404ut_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Let V denote the vector space of all functions ƒ : R → R, equipped with addition + : V × V → V defined
via (f+g)(x) = f(x) + g(x), x ≤ R, and scalar multiplication : R x V → V defined via (\ · ƒ)(x) = Xƒ(x),
xER".
Now let W =
{ƒ: R→R : f(x) = ax + b for some a, b ≤ R}, i.e. the space of all linear functions R → R.
(a) Show that W is a subspace of V. (You may assume that V is a vector space).
[3]
(b) Find a basis for W. You should prove that it is indeed a basis.
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