1. ## space vector

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} , $(x1,y1,z1)+(x2,y2.z2)=(x2,y1+y2,z2)$ and $k(x,y,z)=(kx,ky,kz)$

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

f) $(\alpha + \beta ) * u = \alpha * u + \beta * u$
...
= $(\beta x1, \alpha y1 + \beta y1, \beta z1)$ Not Check

g) 1 * u = u
...
= (x1,y1,z1) Check

h) $\alpha * (u + v) = \alpha * u + \alpha * v$
...
= $(\alpha x2, \alpha y1 + \alpha y2, \alpha z2)$ Not Check

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

a) Check
b) Check
c) Check
d) I could not resolve
e, f , g and h ) Not yet done

Thanks

2. There is no definition of the product of two elements in $V$?!
You put a product with a scalar and an element of $V$, but not between $2$ elements of $V$. Are you sure no information about the problem is missing?
Anyway, I didn't check all but for the 1) $(x1,y1,z1)+(x2,y2.z2)\neq (x2,y2,z2)+(x1,y1,z1)$ so $V$ is not a vector space.
(Detail: $(x1,y1,z1)+(x2,y2,z2)=(x2,y1+y2,z2)$ while $(x2,y2,z2)+(x1,y1,z1)=x1,y2+y1,z1$).

And about the 2), it seems to be a vector space but as you didn't define the product of two elements in V, we cannot affirm it.