Prove that linear maps on $x\in\mathbb{R}^n$ are differentiable

Question

Given $M$ a real $n$-by-$n$ matrix and $w\in \mathbb{R}^n$, use the definition of differentiability to show that $F(x)=Mx+w$ is differentiable at every $x \in \mathbb{R}^n$ with $DF(x) = M$.

This is the very new concept for me so I don't really know how to start. I know what the definition of differentiability is but not sure how to properly use it in this question.

Answer 1

Apply the definition. Show that, for each fixed $x\in\mathbb R^n$ we have

$$\lim_{h\to0} \frac{\lVert F(x+h)-F(x)-Mh\rVert}{\lVert h\rVert} = 0$$