Endomorphism/Invariant linear subspace/Decomposition/Matrix/Exercise

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping on the finite-dimensional ${}K$ -vector space ${}V$ . Let ${}k\leq n$ . Show that there exists a direct sum decomposition ${}V=U\oplus W$ into invariant linear subspaces ${}U,W\subseteq V$ of dimension ${}k$ and ${}n-k$ , if and only if there exists a basis of ${}V$ such that the describing matrix of ${}\varphi$ with respect to this basis has the form

{\begin{pmatrix}a_{11}&\ldots &a_{1k}&0&\ldots &0\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{k1}&\ldots &a_{kk}&0&\ldots &0\\0&\ldots &0&a_{k+1k+1}&\ldots &a_{k+1n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\0&\ldots &0&a_{nk+1}&\ldots &a_{nn}\end{pmatrix}}.