Find a vector that spans the given set

Question

• Question in book:

Let H be the set of all vectors of the form [-2t, 5t, 3t]. Find a vector v in R3 such that H=Span{v}. Why does this show that H is a subspace of R3?

• Answer from solution manual: The set H=Span{v}, where v=[1,3,2]. Thus H is a subspace of R3 by Theorem 1*.

• My question: I know this is stupid but I don't see where the answer v=[1,3,2] comes from. I answered this question as [-2,5,3]. If someone would give explanation as to how this answer came about, I'd be very appreciative!

*Theorem 1: If v_1,…,v_p are in vector space V, then Span{v_1,…,v_p} is a subspace of V.

Answer 1

I think the correct answer is $v=[-2,5,3]$ so then $Span{(v)}=Span{([-2,5,3])}=H$. Because $v\subset \mathbb{R}^3$ so $H$ is a subspace of $\mathbb{R}^3$.

Answer 2

You're answer is correct. From the definition of "span" a base from H would be [-2,5,3]. [1,3,2] is not even a vector from H.