Consider the following sets with operations: ) V = {(r, y) E R² | æ – y = 1}, equipped with the usual vector operations on R². (i) W = {(x, y, z) E R* | 2, y, z are rational}, equipped with the usual vector operations on IR (ii) R? with addition defined by (æ, y) (x', y') = (r- a', y - y') and the usual multiplication by %3D scalars. Choose the correct statement below: A. (1), (ii) and (ii) are vector spaces over R. B. Only (i) and (ii) are vector spaces over R. C. Only (ii) and (iii) are vector spaces over R. D. Only (i) and (iii) are vector spaces over IR. E. Only (i) is a vector space over IR. F. None of (i), (ii) and (iii) are vector spaces over IR.
Consider the following sets with operations: ) V = {(r, y) E R² | æ – y = 1}, equipped with the usual vector operations on R². (i) W = {(x, y, z) E R* | 2, y, z are rational}, equipped with the usual vector operations on IR (ii) R? with addition defined by (æ, y) (x', y') = (r- a', y - y') and the usual multiplication by %3D scalars. Choose the correct statement below: A. (1), (ii) and (ii) are vector spaces over R. B. Only (i) and (ii) are vector spaces over R. C. Only (ii) and (iii) are vector spaces over R. D. Only (i) and (iii) are vector spaces over IR. E. Only (i) is a vector space over IR. F. None of (i), (ii) and (iii) are vector spaces over IR.
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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Question
![Consider the following sets with operations:
(1) V = {(r, y) E R? | r – y = 1} , equipped with the usual vector operations on R?.
(ii) W = {(x, y, z) E R | a, y, z are rational}, equipped with the usual vector operations on R.
(iii) R? with addition defined by (x, y) (x', y') = (x - x', y – y') and the usual multiplication by
scalars.
Choose the correct statement below:
A. (1), (ii) and (i) are vector spaces over R.
B. Only (i) and (ii) are vector spaces over R.
C. Only (ii) and (iii) are vector spaces over R.
D. Only (i) and (iii) are vector spaces over R.
E. Only (i) is a vector space over R.
F. None of (i), (ii) and (iii) are vector spaces over R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F28e6ece8-c0e3-44e8-ab8e-27b7c9c4111b%2F34453363-bf7b-4314-90df-7d69ed4df6c5%2Fn7udfl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the following sets with operations:
(1) V = {(r, y) E R? | r – y = 1} , equipped with the usual vector operations on R?.
(ii) W = {(x, y, z) E R | a, y, z are rational}, equipped with the usual vector operations on R.
(iii) R? with addition defined by (x, y) (x', y') = (x - x', y – y') and the usual multiplication by
scalars.
Choose the correct statement below:
A. (1), (ii) and (i) are vector spaces over R.
B. Only (i) and (ii) are vector spaces over R.
C. Only (ii) and (iii) are vector spaces over R.
D. Only (i) and (iii) are vector spaces over R.
E. Only (i) is a vector space over R.
F. None of (i), (ii) and (iii) are vector spaces over R.
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