Let V = R³ and let H be the subset of V of all points on the plane -3x + 4y + 5z = −12. Is H a subspace of the vector space V?
Let V = R³ and let H be the subset of V of all points on the plane -3x + 4y + 5z = −12. Is H a subspace of the vector space V?
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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![Let V = R³ and let H be the subset of V of all points on the plane −3x + 4y + 5z = −12. ls H a subspace of the vector space V?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92856043-199e-49b6-95ce-27336f8d0cbd%2F19892295-8b4b-4c87-9a1b-1ad55a32f0ff%2Fjnn0i78_processed.png&w=3840&q=75)
Transcribed Image Text:Let V = R³ and let H be the subset of V of all points on the plane −3x + 4y + 5z = −12. ls H a subspace of the vector space V?
![1. Does H contain the zero vector of V?
choose
2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and syntax such as
<1,2,3>, <4,5,6>.
3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a vector in H whose product is not in H, using a comma separated list and
syntax such as 2, <3,4,5>.
4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3.
choose](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92856043-199e-49b6-95ce-27336f8d0cbd%2F19892295-8b4b-4c87-9a1b-1ad55a32f0ff%2F9z5cexs_processed.png&w=3840&q=75)
Transcribed Image Text:1. Does H contain the zero vector of V?
choose
2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and syntax such as
<1,2,3>, <4,5,6>.
3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a vector in H whose product is not in H, using a comma separated list and
syntax such as 2, <3,4,5>.
4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3.
choose
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