2 True or False: Let U and V be distinct, non-trivial subspaces of R°, which are themselves not equal to R. If Sı = U + {v} and S2 = V+ {w} for some v, w E R³ (i.e. these sets are shifts of U and V respectively), then the set S = S1 n S2 = {x |x € S1 and x € S2} is² the shift of some subspace W of R.
2 True or False: Let U and V be distinct, non-trivial subspaces of R°, which are themselves not equal to R. If Sı = U + {v} and S2 = V+ {w} for some v, w E R³ (i.e. these sets are shifts of U and V respectively), then the set S = S1 n S2 = {x |x € S1 and x € S2} is² the shift of some subspace W of R.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
PLEASE SOLVE NO.2, IF POSSIBLE CAN YOU ALSO SOLVE N0.1
![Read the following definitions carefully, and then answer the questions below.
Given a subspace U of R", and a non-zero v E R" we define the following new
subset S of R":
S = U + {v} = {x+v|x € U}
We
say
that S is a shift of U (or that S is the shift of U by v).
1 Find an infinite subset X of R3 which is not a subspace, but is also not the shift of
any subspace of R°. Justify your answer by showing that X is not a subspace, and
that it could not be the shift of any subspace.
2 True or False: Let U and V be distinct, non-trivial subspaces of R³, which are
themselves not equal to R³. If S1 = U+ {v} and S2 = V+ {w} for some v, w E R3
(i.e. these sets are shifts of U and V respectively), then the set S = S1 n S2
x x E S1 and x E S2} is² the shift of some subspace W of R°.
Definition](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbaf174e5-b22e-49e3-bb13-33053eb86002%2Fb5e81a5b-0af6-498d-a118-8b479261d88c%2Fyayyp8o_processed.png&w=3840&q=75)
Transcribed Image Text:Read the following definitions carefully, and then answer the questions below.
Given a subspace U of R", and a non-zero v E R" we define the following new
subset S of R":
S = U + {v} = {x+v|x € U}
We
say
that S is a shift of U (or that S is the shift of U by v).
1 Find an infinite subset X of R3 which is not a subspace, but is also not the shift of
any subspace of R°. Justify your answer by showing that X is not a subspace, and
that it could not be the shift of any subspace.
2 True or False: Let U and V be distinct, non-trivial subspaces of R³, which are
themselves not equal to R³. If S1 = U+ {v} and S2 = V+ {w} for some v, w E R3
(i.e. these sets are shifts of U and V respectively), then the set S = S1 n S2
x x E S1 and x E S2} is² the shift of some subspace W of R°.
Definition
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