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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)
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,...
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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
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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