Determine whether the set is a subspace of R3 under addition

Determine whether the set is a subspace of R³ under addition and scalar multiplication defined on R³.

Solution

1.

Let, (x,y,z) and (p,q,r) be in this set

(x,y,z)+(p,q,r)=(x+p,y+q,z+r)

(x+p)-4(y+q)-(z+r)=(x-4y-z)+(p-4q-r)=0+0=0

Hence, set is closed under addition

2.

Let, (x,y,z) be in this set and c be a scalar

c(x,y,z)=(cx,cy,cz)

(cx)-4(cy)-(cz)=c(x-4y-z)=c*0=0

Hence closed under multiplication

Hence the set is a subspace

Hence, set is closed under addition