The set R2 with the operations x1y1x2y2x1x2y1y2 and cxycxy I

Solution

One way to understand this problem is that we know that for any vector P, 0P = 0. But
we can see that :
(x1,y1)+(x2,y2)=(x1+x2,y1+y2)

is regular addition for vectors, the zero vector then becomes 0 = (0; 0).
But , we have 0(P1; P2) = (0; P2) as well and this is not the zero vector for P2 not equal to 0.
Thus we cannot have a vector space.
We can check this using the axion and we\'ll cone to know that the axiom fails :
(a + b)*(x1; x2) = ((a + b)x1; x2); ------> (1)
while
a*(x1; x2) + b*(x1; x2) = (ax1; x2) + (bx1; x2) = ((a + b)x1; 2x2); ----------> (2)
and we can see that (1) and (2) are not the same.

Hence we ccan say that
(x1,y1)+(x2,y2)=(x1+x2,y1+y2) and c(x,y)=(cx,y)
Is not a vector space.