MHB Prove A=B when A⊂span(B) and B⊂span(A)

[toni07]

Let A and B be subsets of a vector space V. Assume that A ⊂ span(B) and that B ⊂ span(A) Prove that A = B.
I don't know how to go about this question, any help would be appreciated.

 

[Fernando Revilla]

Let A and B be subsets of a vector space V. Assume that A ⊂ span(B) and that B ⊂ span(A) Prove that A = B.

That is not true. Choose for example $V=\mathbb{R}^2,$ $A=\{(1,0)\}$ and $B=\{(2,0)\}.$

 

[caffeinemachine]

Let A and B be subsets of a vector space V. Assume that A ⊂ span(B) and that B ⊂ span(A) Prove that A = B.
I don't know how to go about this question, any help would be appreciated.

Quite likely what you were actually after is the following:

If $A\subseteq\text{ span}(B)$ and $B\subseteq\text{ span}(A)$, then $\text{span}(A)=\text{span}(B)$.

Let $\text{span}(A)=U$ and $\text{span}(B)=W$.

Since $A\subseteq W$, we have $\text{span}(A)=U\subseteq W$. This is because $W$ is a subspace of $V$. Similarly $W\subseteq U$. We get $W=U$ and we are done.