Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I want to show that if for $f: \mathbb{R}^2 \mapsto \mathbb{R}^2$ if we have

$f\begin{pmatrix} a_1 + v_1 \\ a_2 + v_2 \\ \end{pmatrix}$ = $f\begin{pmatrix} a_1 \\ a_2 \\ \end{pmatrix} + [Df(a)] \large\vec{v}$

for all $a,v \in \mathbb{R}^2$

then $f$ is linear.

Obviously the derivative is linear, not sure how that can help me here though, since I can't see any immediate use for the equality $[Df(a)] \large\vec{v}$ $=$ $[Df(a\vec{ v})]$

share|cite|improve this question
One approximates a differentiable function by a linear function; what happens if that function equals this approximation? Answer: It is linear. :) Hint: Try to verify the linearity axioms. – awllower Mar 10 '14 at 13:36
Well, that's akin to showing $D[f(a)]\vec{v} = f(\begin{matrix} v_1 \\ v_2 \end{matrix}$ And to be honest, I'm not sure how to do that without f being linear.. – terrible at math Mar 10 '14 at 19:22
in the first line there may be a typo: should be $\to$ rather than $\mapsto$ – rych Mar 11 '14 at 7:41

1 Answer 1

You have $\forall x\in\mathbb{R}^2,f(x)=f(0)+Df(0)x$ which looks linear to me.

share|cite|improve this answer
is showing only 0 work slike that sufficient to show its linear? – terrible at math Mar 11 '14 at 19:35

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.