Let V = span{$v_1$,...,$v_k$}. Prove that {$v_1$,...,$v_k$} is a minimal spanning set of V if and only if $v_1$,...,$v_k$ are linearly independent.

I am trying to do this by proving the contrapositive, i.e. $v_1$,...,$v_k$ are linearly dependent if and only if {$v_1$,...,$v_k$} is not a minimal spanning set of V.

Linear dependence implies there exists $c_1$,...,$c_k$, not all zero, such that $c_1$$v_1$+...+$c_k$$v_k$ = $0$

How can I use this to prove that {$v_1$,...,$v_k$} is not a minimal spanning set of V?


If $B=\{v_1,\dots,v_k\}$ is not a minimal spanning set, then we can remove WLOG $v_k$ and $\{v_1,\dots,v_{k-1}\}$ is still a spanning set. We may assume $v_k\neq 0$, otherwise the original set was clearly not linearly independent. So $v_k=c_1v_1+\cdots+c_{k-1}v_{k-1}$ and the original set was not linearly independent.

Conversely, if $v_i$ are linearly dependent, then WLOG assume $v_k$ is in the span of $\{v_1,\dots,v_{k-1}\}$. If $B$ was not a spanning set we are done. If $B$ was a spanning set, then it cannot be minimal because $v_k\in\text{Span}\{v_1,\dots,v_{k-1}\} \implies \{v_1,\dots,v_{k-1}\}\;\;\text{is a spanning set}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.