Please show all steps and operations clearly please Is the s

Please show all steps and operations clearly please.

Is the set U 3{ y, z) E R3 z y 32 0 a subspace? If the answer is yes, what is the dimension of this subspace? (r,

1. (0,0,0) belong to this set

2. Let, (x,y,z) and (r,s,t) be in this set

(x,y,z)+(r,s,t)=(x+r,y+s,z+t)

(x+r)+(y+s)+3(z+t)=x+y+3z+r+s+3t=0+0=0

Hence it is a subspace

x+y+3z =0 ie x=-y-3z

So general vector is

(x,y,z)=(-y-3z,y,z)=y(-1,1,0)+z(-3,0,1)

HEnce basis is {(-1,1,0),(-3,0,1)}

Hence dimension is 2