basis is minimal spanning set

Question

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 ?

Answer 1

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.