Suppose that $f: \mathbb R^q \to \mathbb R^p$ is a linear map. Prove that $f$ is differentiable and that $f'(x) = f$ for every $x \in \mathbb R^q$

Suppose that $f: \mathbb R^q \to \mathbb R^p$ is a linear map. Prove that $f$ is differentiable and that $f'(x) = f$ for every $x \in \mathbb R^q$.

I don't know of any way to prove this?

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To say that $f$ is differentiable at $x$ means there exists a linear transformation $T$ such that $$f(x + \Delta x) = f(x) + T(\Delta x) + o(\Delta x)$$ as $\| \Delta x \| \to 0$.
In your problem, there does exist such a linear transformation $T$, namely $T = f$. Indeed, $$f(x + \Delta x) = f(x) + f(\Delta x) + 0.$$
Hence $f$ is differentiable at $x$ and $f'(x) = T$.