Vector space/Finite family/Linearly independent and dimension of spanned subspace/Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Let ${}v_{1},\ldots ,v_{m}$ be a family of vectors in ${}V$, and let

Failed to parse (syntax error): {\displaystyle {{}} U = \langle v_i ,\, i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Vector space/Finite family/Linearly independent and dimension of spanned subspace/Exercise]] __NOINDEX__ 1 , \ldots , m \rangle \, }

be the linear subspace

they span. Prove that the family is linearly independent if and only if the dimension of ${}U$ is exactly ${}m$.

Create a solution