Vector space/n vectors/Equivalence relation by same linear subspace/Exercise

Let ${}K$ be a field, and let ${}V$ denote a ${}K$-vector space, and ${}m\in \mathbb {N}$. We consider on the product set ${}V^{m}$ the following relation.

$(v_{1},\ldots ,v_{m})\sim (w_{1},\ldots ,w_{m}){\text{ if }}\langle v_{1},\ldots ,v_{m}\rangle =\langle w_{1},\ldots ,w_{m}\rangle$

Show that this is an equivalence relation. Give an bijection between the corresponding quotient set and the set of linear subspaces of ${}V$ of dimension ${}\leq m$. Moreover, show that two tuples ${}(v_{1},\ldots ,v_{m})$ and ${}(w_{1},\ldots ,w_{m})$ are equivalent in this relation if and only if there exists an invertible ${}m\times m$-matrix ${}M={\left(a_{ij}\right)}_{ij}\in \operatorname {Mat} _{m}(K)$ such that

${}v_{i}=\sum _{j=1}^{m}a_{ij}w_{j}\,$

holds for all ${}i$.

Create a solution