Which of the following sets are subspaces of r3

Question

Can somebody please help me with this question, I've tried using these steps 1) {(0,0,0)} 2) Line through the origin. 3) Plane through the origin. but I must have done something wrong. thank you Which of the following sets are subspaces of R3

A. {(x,x+7,x+3) | x arbitrary number }

B. {(x,y,z) | 7x+3y+2z=0}

C. {(x,y,z) | x,y,z>0}

D. {(x,y,z) | −6x−5y−4z=9}

E. {(x,0,0) | x arbitrary number }

F. {(−9x,6x,5x) | x arbitrary number }

Answer 1

A) is not a subspace because it does not contain the zero vector

B) is a subspace (plane containing the origin with normal vector (7, 3, 2)

C) is not a subspace... does not contain the zero vector, and negative scalar multiples of elements of this set lie outside the set.

D) is not a subspace... it's a plane, but it does not contain the zero vector.

E) is a subspace (it's a line through the origin)

F) is a subspace (also a line through the origin)

Note: I'm reading E) to mean all vectors of the form (x, 0, 0) where x ∈ R and similarly for F)