# Span of vector spaces

#### crypt50

##### New member
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.

#### Fernando Revilla

##### Well-known member
MHB Math Helper
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

##### Well-known member
MHB Math Scholar
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.
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.