Endomorphism/Invariant linear subspace/Matrix/Exercise

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional ${}K$ -vector space ${}V$ . Let ${}k\leq n$ . Show that there exists an invariant linear subspace ${}U\subseteq V$ of dimension ${}k$ , 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}&a_{1k+1}&\ldots &a_{1n}\\\vdots &\vdots &\vdots &\vdots &\vdots &\vdots \\a_{k1}&\ldots &a_{kk}&a_{kk+1}&\ldots &a_{kn}\\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}}.