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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb41de797-8c36-43f3-a49e-0d77bbbd163e%2F9803bf8b-7139-4c90-92c5-3e18116b774b%2Fmlxs419_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 (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).
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)