## rosy 3 years ago can any one tell that in vector space spanning set is the smallest subspace of vector space or not?

1. Walleye

Im not exactly sure what you are asking. The smallest subspace is always 0

2. Walleye

You may be wondering if a set of vectors that spans a vector space is the smallest number of vectors to form a basis then no think of a set of vectors that spans 2 space (0,1) and (1,0) and (1,1) This is will span R^2 but you dont "need (1,1) because it is linearly dependent of (0,1) and (1,0)

3. Walleye

If this is what you are asking then take a look at contraction and expansion thms. These should lead you to the answer

4. rosy

actually my que is the part of a theorem THEOREM:Let v1,v2,v3...,vn be vectors in a vector space V and let their span be W=span{v1,v2,....,vn} then, (a) W is a subspace of V. (b) W is the smallest subspace of V that contains all of the vectors . and i want the proof of its 2nd part. can u help me plz....

5. Walleye

Yes I see what you are saying now. Try proof by contridiction. That is, assume that W is not the smallest subspace

6. rosy

i tried but do not get the point plz u proof it if u can...

7. Walleye

This proof is kind of tricky just because it is so trival. Im not going to give you the answer but here is how i would go about the proof without using contridiciton. If you wanted to make the smallest subspace you would just put in the vectors v1, .... , vn

8. Walleye

but subspaces force you to be able to add and multiply any two vectors and remain in the subpsace

9. Walleye

Therefore if you want a subspace with all of the vectors you must have the span of all of the vectors

10. Walleye

Basically, say what the smallest subspace can be (just all of the vectors by themselfs) then say the def of a subspace forces the span of all of the vectors to be the smallest subspace

11. rosy

from all these how can u say w is SMALLEST subspace?

12. Walleye

ok ok Ill do the proof :)

13. Walleye

The smallest SET that contains v1, ....,vn is (v1,....vn) Now is this SET a SUBSPACE? no not nessarly can we take any vi and any vj and have vi + vj is an element of our SET so we need to have any vi + vj to be in our SET and any constant,c, cvi to make it a SUBSPACE. the span of v1,...,vn by def is a SET that contains v1,...,vn and any vi + vj and cvi so this span is the smallest SUBSPACE becasue it contains no more than v1,...,vn, and has closure under additon and scalar mutiplication

14. Walleye

I uppercased set and subspace to make sure you didnt confuse them as you were reading it.

15. Walleye

now the last part where i claim span(v1, ... vn) is the smallest should be ok for most poeple. if you want to be super rigourous then say, make the subspace any smaller and it will not contain every c*(vi + vj) and thus making it not a subspace

16. Walleye

no not nessarly can we take any vi and any vj and have vi + vj NOT an element of our SET

17. rosy

thanks a lot. now i try to understand it

18. Walleye

ok good! just remeber this... start with the smallest set of v1,....vn see if your set is a subspace add enough vectors to make a subspace show that if you take away any vectors you will lose closure under addition or multiplication