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3. Let V denote the vector space of all functions f: Rn R, equipped with addition + : V × V → V defined via (f+g)(x) = f(x)+g(x),x E R", and scalar multiplication RX V → V defined via (A. f)(x) = \f(x), xER". Now let W = {ƒ: R → R:f(x) = ax +b for some a, b e R}, i.e. the space of all linear functions R" → R. (a) 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: Rn R, equipped with addition + : V × V → V defined via (f+g)(x) = f(x)+g(x),x E R", and scalar multiplication RX V → V defined via (A. f)(x) = \f(x), xER". Now let W = {ƒ: R → R:f(x) = ax +b for some a, b e R}, i.e. the space of all linear functions R" → R. (a) Find a basis for W. You should prove that it is indeed a basis.

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 f : R" → R, equipped with addition + : V × V → V defined via (ƒ+g)(x) = f(x)+ g(x), x ≤ R", and scalar multiplication : R × V → V defined via (\ · ƒ)(x) = \ƒ(x), ● x ER". Rº Now let W = {ƒ : R^ → R : ƒ(x) = ax + b for some a, b € R}, i.e. the space of all linear functions R" → R. (a) Find a basis for W. You should prove that it is indeed a basis.
Transcribed Image Text:3. Let V denote the vector space of all functions f : R" → R, equipped with addition + : V × V → V defined via (ƒ+g)(x) = f(x)+ g(x), x ≤ R", and scalar multiplication : R × V → V defined via (\ · ƒ)(x) = \ƒ(x), ● x ER". Rº Now let W = {ƒ : R^ → R : ƒ(x) = ax + b for some a, b € R}, i.e. the space of all linear functions R" → R. (a) Find a basis for W. You should prove that it is indeed a basis.
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