Question on Remark in Axler's Linear Algebra

[Group_Complex]

Hello, i am studying vector subspacess and Axler introduces the two criteria for a vector subspace (closure under addition and scalar multiplication).
He then proceeds to give an example; (x1,x2,x3,x4) belonging to F^4 : x3=x4+b, where b is an element of F. Axler states that this example is not a subspace unless b=0, yet this is the same space as V and i was under the impression (Axler states it himself) that V is a subspace of itself? Should not any value of b in F be possible?

 

[Group_Complex]

I think i have realized where i went wrong. The zero vector must be contained within the subspace, thus b=0 is the only solution which allows this. Is that a suitable method to complete the example or am i missing something else?

 

[lavinia]

I think i have realized where i went wrong. The zero vector must be contained within the subspace, thus b=0 is the only solution which allows this. Is that a suitable method to complete the example or am i missing something else?

that is correct.