Thread: Is this a vector space?

1. Is this a vector space?

Let (a,b) and (c,d) be in V and c be in R, we define:
(a,b) + (c,d) = (a+2c,b+3d) and c(a,b) = (ca,cb)

Using the eight properties.
I know that the multiplication is normal, so those properties work, but the addition does not hold under the conditions, does it?

(u + v) + w =/= u + ( v + w)
u + v =/= v + u
-> If u = (a,b) and v = (c,d),
u + v = (a,b) + (c,d) = (a+2c,b+3d)
v + u = (c,d) + (a,b) = (c+2a,d+3b)

So it's not a vector space. Is this correct ??

2. Originally Posted by Volcanicrain
Let (a,b) and (c,d) be in V and c be in R, we define:
(a,b) + (c,d) = (a+2c,b+3d) and c(a,b) = (ca,cb)

Using the eight properties.
I know that the multiplication is normal, so those properties work, but the addition does not hold under the conditions, does it?

(u + v) + w =/= u + ( v + w)
You should show specifically: if u= (a,b), v= (c,d), and w= (e, f) then u+ v= (a+2c, b+3d) and then (u+ v)+ w= (a+2c+ 2u, b+3d+ 3f) while v+w= (c+2e, d+ 3f) and u+ (v+w)= (a+ 2(c+3e),b+3(d+3f))= (a+2c+ 6e, b+3d+ 9f) which is NOT the same as (a+2c+2u, b+3d+3f).

u + v =/= v + u
-> If u = (a,b) and v = (c,d),
u + v = (a,b) + (c,d) = (a+2c,b+3d)
v + u = (c,d) + (a,b) = (c+2a,d+3b)
Okay, here you did exactly that.

So it's not a vector space. Is this correct ??
Yes. In fact, since all properties have to be true you really only needed to show that $u+ v\ne v+ u$.

3. Thank you very much!
I did the (u+v)+w = u+(v+w) out on paper as you did only to realize the other one was shorter haha.
Thank you very much for answering my question, you are very clear