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- Thread starter crypt50
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- Jan 29, 2012

- 661

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

- Mar 10, 2012

- 834

Quite likely what you were actually after is the following: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.

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.