Determine whether data sets, operations, are vector spaces.

[Apprentice123]

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
(x1,y1,z1)+(a,b,c)=(x1,y1,z1)
a=x1
b=0
c=z1
...
u+e=u Check

d) u+(u')=e
(x1,y1,z1)+(d,e,f)=(x1,0,z1)
d=x1
e=-y1
f=z1
....
u+(u')=e Check

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) Check
e) Check
f) check
g) check
h) check


Please, the first is correct?