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

${}U=\langle v_{i},\,i=1,\ldots ,m\rangle \,$

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