Vector space/Finite family/Properties of substitution/Exercise

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

$\varphi \colon K^{n}\longrightarrow V,(s_{1},\ldots ,s_{n})\longmapsto \sum _{i=1}^{n}s_{i}v_{i},$

and prove the following statements.

1. ${}\varphi$ is injective if and only if ${}v_{1},\ldots ,v_{n}$ are linearly independent.
2. ${}\varphi$ is surjective if and only if ${}v_{1},\ldots ,v_{n}$ is a system of generators for ${}V$.
3. ${}\varphi$ is bijective if and only if ${}v_{1},\ldots ,v_{n}$ form a basis.

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