Let V be the set of ordered pairs (a,b) of real numbers. Showthat V is not a vec

Question

Let V be the set of ordered pairs (a,b) of real numbers. Showthat V is not a vector space over R with addition andscalar multiplication defined by: (a,b) + (c,d) = (ac,bd) and k(a,b)=(ka,kb) Hi i was wondering if this is a correct way of disproving thisbeing a vector space. Using the fact that v + 0 = 0 + v = v (1,2) + (0,0) = (0,0) which does notequal (1,2) v in this case. Let V be the set of ordered pairs (a,b) of real numbers. Showthat V is not a vector space over R with addition andscalar multiplication defined by: (a,b) + (c,d) = (ac,bd) and k(a,b)=(ka,kb) Hi i was wondering if this is a correct way of disproving thisbeing a vector space. Using the fact that v + 0 = 0 + v = v (1,2) + (0,0) = (0,0) which does notequal (1,2) v in this case.

Explanation / Answer

If (a,b)+(c,d)=(ac,bd) we have a case where k=(0,1) suchthat:
(1,0)+k=(0,0)+k Clearly (1,0) does not equal(0,0)
There is a logical contradiction so this proves that V is nota vector space over R with that definition of addition.

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