Linear Algebra and Matrices

Question

Let V = {(x, y)|x ∈ R, y ∈ R, y > 0}, define addition and scalar multiplication on V by
(x, y) + (z, w) = (x + z, yw), and a(x, y) = (ax, ya)
Is V a vector space over R with these operations? Justify your answer.

Answer 1

For V to be a vector space, V must be closed under scalar multiplication.

 

That is,  if (x,y) ∈ V and c is a real number (scalar), then c(x,y) ∈ V

 

I will show that the statement above is false for the given set, V:

 

   Let c = -2 and (x,y) = (1, 3).

 

   (1,3) ∈ V, but -2(1,3) = (-2,-6) ∉ V

 

Since the set is not closed under scalar multiplication, the set is NOT a vector space.  

 

 

Answer 2

Suppose V is a vector space over R with these operations.  Then the following would have to be true:

 

a[(x,y) + (z,w)] = a(x,y) + a(z,w) = (ax,ay) + (az,aw) = (ax+az, a²yw)

 

But it will also have to be true that

 

a[(x,y) + (z,w)] = a(x+z, yw) = (a(x+z), ayw) = (ax+az, ayw)

 

Then ayw would have to equal a²wy for any real a, which is not true in general.  Therefore, V is not a vector space over R with these operations.