Linear mapping/To finite-dimensional/Representation with linear forms/Fact

Let ${}K$ be a field and let ${}V$ and ${}W$ be vector spaces over ${}K$, where ${}W$ is finite-dimensional. Let

$\varphi \colon V\longrightarrow W$

denote a linear mapping.

Then there exist vectors

${}w_{1},\ldots ,w_{n}\in W$ and linear forms ${}f_{1},\ldots ,f_{n}$ on ${}V$ such that

${}\varphi =f_{1}w_{1}+f_{2}w_{2}+\cdots +f_{n}w_{n}\,$

holds.

Proof, Write another proof