Do these sets span the same space?

[Bipolarity]

Suppose you have two sets $S_{1}$ and $S_{2}$. Suppose you also know that every vector in $S_{1}$ is expressible as a linear combination of the vectors in $S_{2}$. Then can you conclude that the two sets span the same space?

If not, what if you further knew that every vector in $S_{2}$ is expressible as a linear combination of the vectors in $S_{1}$?

I merely need an answer. I will work out the details (proof) for myself. Thanks!

BiP

 

[jedishrfu]

Isn't this like if every element of set A is in set B then there may still be elements in B that aren't in A.

Geometrically if S1 is all vectors in a plane whereas S2 is all vectors in a 3D space then its clear that S2 has some vectors that aren't in S1.

 

[micromass]

Suppose you have two sets $S_{1}$ and $S_{2}$. Suppose you also know that every vector in $S_{1}$ is expressible as a linear combination of the vectors in $S_{2}$. Then can you conclude that the two sets span the same space?

If not, what if you further knew that every vector in $S_{2}$ is expressible as a linear combination of the vectors in $S_{1}$?

Yes, if you know both those things, then you can make the conclusion. If you only know one of these things, then jedishrfu already pointed out that the conclusion doesn't follow.

 

[WannabeNewton]

No to the first and yes to the second. For the first one you can come up with tons of counter-examples. The second one you can prove straightforwardly from the definition of span.

 

[HallsofIvy]

If every vector in S2 can be written as a linear combination of vectors in S1, then S2 spans some subspace of the span of S1.