Which of the following sets are subspaces of R3 SolutionA C

Which of the following sets are subspaces of R³ ?

A, C and E are subspaces ,as they are defined by homogeneous equations

(or directly check that the solution sets are closed under addition and scalar multiplication)

For example,C:

If (x,y,z) and (u,v,w) satisfy

5x+9y=0 , -4x-6z=0

5u+9v=0, -4u-6w=0

then (x+u,y+v,z+w) is also in the solution set:

5(x+u)+9(y+v)=0 and -4(x+u)-6(z+w)=0

and clearly (kx+ku,ky+kv,kz+kw) is also in the solution set.

B is not a subspace--for instance, (0,0,0) is not in the solution set.(any subspace must contain the origin)

D is not a subspace--same reason as above.

F is not a subspace--again (0,0,0) does not belong to the solution set