Given u(x,y,z),
(x1,y1,z1)+(x2,y2,z2) = (x1+x2,y1+y2,3)
s(x,y,z)=(sx,sy,3)
Assuming vector space is defined over field F such as R.
(k+m)(u)
=(k+m)(x,y,3) [definition of u]
=((k+m)x,(k+m)y,3) [definition of scalar multiplication]
=(kx+mx, ky+my,3) [distributivity defined over F]
=(kx,ky,3)+(mx,my,3) [definition of addition]
=k(x,y,3)+m(x,y,3) [definition of scalar multiplication]
=ku+mu [definition of u] QED