Let ? and ?? denote two distinct objects, neither of which is in ????. Define
an addition and scalar multiplication on ???? ? {?, ??} as you could guess
from the notation. Specifically, the sum and product of two real numbers is
as usual, and for ???? ? ???? define
????? =
?{{?{{?
?? if ???? < 0,
0 if ???? = 0,
? if ???? > 0,
????(??) =
?{{?{{?
? if ???? < 0,
0 if ???? = 0,
?? if ???? > 0,
???? + ? = ? + ???? = ? + ? = ?,
???? + (??) = (??) + ???? = (??) + (??) = ??,
? + (??) = (??) + ? = 0.
With these operations of addition and scalar multiplication, is ???? ? {?, ??}
a vector space over ????? Explain