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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 (X. f)(x) = \f(x), xER". Now let W = {f: R" → R: f(x) = ax + b for some a, b E 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. 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 (X. f)(x) = \f(x), xER". Now let W = {f: R" → R: f(x) = ax + b for some a, b E 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).

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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3. Let V denote the vector space of all functions ƒ : R" → R, equipped with addition +: V × V → V defined via (ƒ+g)(x) = ƒ(x)+g(x), x = R", and scalar multiplication · : R × V → V defined via (λ · ƒ)(x) = \ƒ(x), x ER". Now let W = {ƒ {f: : Rª → R : ƒ(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).
Transcribed Image Text:3. Let V denote the vector space of all functions ƒ : R" → R, equipped with addition +: V × V → V defined via (ƒ+g)(x) = ƒ(x)+g(x), x = R", and scalar multiplication · : R × V → V defined via (λ · ƒ)(x) = \ƒ(x), x ER". Now let W = {ƒ {f: : Rª → R : ƒ(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).
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