Vector space/Dimension n and n vectors/Equivalences/Fact/Proof/Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space of dimension ${}n=\dim _{K}{\left(V\right)}$. Suppose that ${}n$ vectors ${}v_{1},\ldots ,v_{n}$ in ${}V$ are given. Prove that the following facts are equivalent.

1. ${}v_{1},\ldots ,v_{n}$ form a basis for ${}V$.
2. ${}v_{1},\ldots ,v_{n}$ form a system of generators for ${}V$.
3. ${}v_{1},\ldots ,v_{n}$ are linearly independent.

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