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trixitium
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Homework Statement
Determine if the following set is a vector space under the the given operations.
The set V of all pairs of real numbers of the form (1,x) with the operations:
(1, y) + (1, y') = (1, y + y')
k(1, y) = (1, ky)
Homework Equations
The Attempt at a Solution
Axiom 4: There is an object 0 in V, called a zero vector for V, such that 0 + u = u + 0 = u.
u = (1, y)
(1, y) + (1, 0) = (1, y+ 0) = (1, u) = u
May I consider (1,0) as the zero vector? I have a doubt if the zero vector 0 has to be (0,0) always or the zero can be defined as any vector (in this case (1,0) ) that remains the vector untouched under addition.
Axim 5: for each u in V, there is an object -u in V, called a negative of u, such that u + (-u) = (-u) + u = 0
u = (1,x)
-u = (-1)(1,x) = (1, -x) by definition of scalar multiplication given for the set V. (Is this correct?)
then u + (-u) = (1,x) + (-1)(1,x) = (1,x) + (1,-x) = (1,x+(-x)) = (1,0) = 0
in this case (1,0) is the zero vector.