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4) Let oo and -0o denote two distinct objects, neither of which is in R. Define an addition and scalar multiplication on RU {c0} U {-0} as follows. The product and sum ot two real numbers as as usual, and for t e R define (-00 too =} 0 if t = 0, o if t<0, t(-00) =} 0 if t = 0, if t>0, if t< 0, 00 if t> 0, (-00 t+0 = 00 +t = 00, t+(-0) = (-00) +t = -0, 00 + 0 = 00, 0 + (-0) = -0 + co = 0. (-00) + (-0) = -0, Is RU{0} U{-0} a vector space over R? Make your to fully justify your answer.

4) Let oo and -0o denote two distinct objects, neither of which is in R. Define an addition and scalar multiplication on RU {c0} U {-0} as follows. The product and sum ot two real numbers as as usual, and for t e R define (-00 too =} 0 if t = 0, o if t<0, t(-00) =} 0 if t = 0, if t>0, if t< 0, 00 if t> 0, (-00 t+0 = 00 +t = 00, t+(-0) = (-00) +t = -0, 00 + 0 = 00, 0 + (-0) = -0 + co = 0. (-00) + (-0) = -0, Is RU{0} U{-0} a vector space over R? Make your to fully justify your answer.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 42E: Rather than use the standard definitions of addition and scalar multiplication in R3, let these two...
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4) Let co and -o denote two distinct objects, neither of which is in R. Define an addition and scalar multiplication on RU {0} U {-0} as follows. The product and sum ot two real numbers as as usual, and for t e R define if t < 0, 0 ift = 0, if t>0, (-00 if t< 0, 00 too ={ 0 if t = 0, t(-00) = if t> 0, t+ 0 = 00 +t = 00, 00 + 00 = 00, t+(-0) = (-00) +t = -o, (-00) + (-0) = -00, 00 + (-00) = -∞ + 00 = 0. Is RU {o0} U{-o} a vector space over R? Make your to fully justify your answer.
Transcribed Image Text:4) Let co and -o denote two distinct objects, neither of which is in R. Define an addition and scalar multiplication on RU {0} U {-0} as follows. The product and sum ot two real numbers as as usual, and for t e R define if t < 0, 0 ift = 0, if t>0, (-00 if t< 0, 00 too ={ 0 if t = 0, t(-00) = if t> 0, t+ 0 = 00 +t = 00, 00 + 00 = 00, t+(-0) = (-00) +t = -o, (-00) + (-0) = -00, 00 + (-00) = -∞ + 00 = 0. Is RU {o0} U{-o} a vector space over R? Make your to fully justify your answer.
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