Hom(V,W)/Several morphisms/Rank estimate/Exercise

Let ${}V$ and ${}W$ denote finite-dimensional ${}K$-vector spaces, and let ${}\varphi _{1},\ldots ,\varphi _{k}\in \operatorname {Hom} _{K}{\left(V,W\right)}$. Show

${}\operatorname {rk} \,{\left(\varphi _{1}+\cdots +\varphi _{k}\right)}\leq \sum _{i=1}^{k}\operatorname {rk} \,\varphi _{i}\,.$

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