Question

list five elements and determine whether the set is a vector space: (verify the 10 axioms):
V={(x,y,z) | z=3}. addition defined by (x1,y1,z1)+(x2,y2,z2) = (x1+x2,y1+y2,3) scalar multiplication defined by s(x,y,z)=(sx,sy,3)

Answer 1

list the ten axioms and check. this just fixes z at 3, so it might as well be
\[V=\mathbb R^2\]

Answer 2

yea im in the third axiom.. u+(v+w)

Answer 3

for example you can check that
\[v_1+v_2=v_2+v_1\]

Answer 4

and i dont know how to add them

Answer 5

ok
\[u+(v+w)=(<x_1,y_1,3>+<x_2,y_2,3>)+<x_3,y_3,3>\]

Answer 6

addition is given by
\[(<x_1,y_1,3>+<x_2,y_2,3>)=<x_1+x_2,y_1+y_2,3>\]

Answer 7

and then
\[(<x_1,y_1,3>+<x_2,y_2,3>)+<x_3,y_3,3>\]
\[=<x_1+x_2,y_1+y_2,3>+<x_3,y_3,3>\]
\[=<x_1+x_2+x_3,y_1+y_2+y_3,3>\]

Answer 8

now do it the other way and see that you get the same thing. that is check that
\[<x_1,y_1,3>+(<x_2,y_2,3>+<x_3,y_3,3>)=<x_1+x_2+x_3,y_1+y_2+y_3,3>\] as well

Answer 9

ohh,,, i was adding lol

=<x1+x2,y1+y2,3>+<x3,y3,3>
=<x1+x2+x3,y1+y2+y3,6>

Answer 10

stupid me... cool cool i get it