Linear mapping/Bijective/Inverse mapping linear/Fact/Proof/Exercise

Let ${}K$ be a field and let ${}V$ and ${}W$ be ${}K$-vector spaces. Let

$\varphi \colon V\longrightarrow W$

be a bijective linear map. Prove that also the inverse map

$\varphi ^{-1}\colon W\longrightarrow V$

is linear.

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