I did the work but not sure if its right, also my professor likes us to include every detail(including all the Vector space Axion) , if there is another way of proving it, more elegant,, please help,, thanks

 

1) Let V be the set of all pairs (x,y) of real numbers with the addition + and scalar multiplication* defined by:

(x1,y1)+(x2,y2)= (x1 + x2 , y1+y2) and c*(x,y)=(x,cy)

 

Show that V with the above operation is not a vector space. Find at least one axiom that fails and give an example showing that the axiom fails..

 

***Let α = (x, y)

Then for real numbers a and b we have

(a + b) α = (a + b) (x, y)

= ( x, (a+b)y )

 

Now aα = a(x, y)

= (x, ay)

bα = b(x, y)

= (x, by)

and aα + bα = (x, ay) + (x, by)

= ( 2x , ay + by)

(a + b) α

Therefore, V is not a vector space.

yeah that violates distributivity, which I believe is an axiom of vector spaces