Welcome to our community

Be a part of something great, join today!

Span of vector spaces

crypt50

New member
Jun 29, 2013
21
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

Well-known member
MHB Math Helper
Jan 29, 2012
661
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
Mar 10, 2012
834
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.