Check whether the data sets together with the operations indicated is a space vector. If not, list the properties that fail

1) V={(x,y,z)/x,y,z E R} , $\displaystyle (x1,y1,z1)+(x2,y2.z2)=(x2,y1+y2,z2)$ and $\displaystyle k(x,y,z)=(kx,ky,kz)$

My Solution:

a) u+v=v+u

(x1,y1,z1)+(x2,y2,z2) = (x2,y1+y2,z2) = y1 + (x2,y2,z2) Not Check

b) u+(v+w) = (u+v)+w

...

= (x3,y1+y2+y3,z3) Not Check

c) u+e=u

I could not resolve

d) u+(u')=e

I could not resolve

e) $\displaystyle \alpha \beta * u= \alpha * (\beta * u)$

...

= $\displaystyle \alpha * [ \beta * (x1,y1,z1)]$ Check

f) $\displaystyle (\alpha + \beta ) * u = \alpha * u + \beta * u$

...

= $\displaystyle (\beta x1, \alpha y1 + \beta y1, \beta z1)$ Not Check

g) 1 * u = u

...

= (x1,y1,z1) Check

h) $\displaystyle \alpha * (u + v) = \alpha * u + \alpha * v$

...

= $\displaystyle (\alpha x2, \alpha y1 + \alpha y2, \alpha z2)$ Not Check

2) V={(x,y)/x,y E R} , $\displaystyle (x1,y1)+(x2,y2)=(x1.x2,y1.y2) $ and $\displaystyle k(x,y)=(kx,ky)$

a) Check

b) Check

c) Check

d) I could not resolve

e, f , g and h ) Not yet done

Thanks