# basis is minimal spanning set

I know minimal spanning set is independent . But how do i prove that a spanning set which is linearly independent i.e a basis is minimal spanning set ? I tried to prove by contradiction by assuming that a set which is smaller and different than the basis set is a minimal set . But this way prove seems to be very lengthy. is it possible that a smaller set containing different independent vectors (other than those is S) is minimal spanning vector ?

• i tried 5 hours to solve this problem . – Flintoff Feb 15 '18 at 17:25
• this is not a homework or assignment problem – Flintoff Feb 15 '18 at 17:25
• Just from curiosity of my mind – Flintoff Feb 15 '18 at 17:26

## 1 Answer

Let $S$ be your basis. If it was not a minimal spanning set, there would be a $s\in S$ such that $\bigl\langle S\setminus\{s\}\bigr\rangle$ is the whole space. Then, $s$ can be written as a linear combination $\alpha_1s_1+\cdots+\alpha_ns_n$ of elements of $S\setminus\{s\}$. That is, we have$$1.s-\alpha_1s_1-\cdots-\alpha_ns_n=0,$$ which is impossible, since $S$ is linearly independent.

• but is it possible that a smaller set containing different independent vectors (other than those is S) is minimal spanning vector ? – Flintoff Feb 15 '18 at 17:39
• The cardinality of all the bases of a vector space is the same, so it is not possible. – Laura Feb 15 '18 at 17:44
• @neraj Are you assuming that $S$ is finite? – José Carlos Santos Feb 15 '18 at 17:44