Vector space/Flag/Bijection/Exercise

Let ${}V$ and ${}W$ be vector spaces over ${}K$ of the same dimension ${}n$, and let

${}0=V_{0}\subset V_{1}\subset \ldots \subset V_{n-1}\subset V_{n}=V\,$

and

${}0=W_{0}\subset W_{1}\subset \ldots \subset W_{n-1}\subset W_{n}=W\,$

be flags in ${}V$ and in ${}W$, respectively. Show that there exists a bijective linear mapping

$\varphi \colon V\longrightarrow W$

such that

${}\varphi (V_{i})=W_{i}\,$

for all ${}i=0,1,\ldots ,n$.

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