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The teacher's notes say that $Df(\xi)(v) = D_vf(\xi) = A(v)$ for some linear function $A$, which I understand. But then he carries on to say that the directional derivative $D_vf(\xi)$ is linear dependent on $v$. Can someone shed some light on this? I know that two vectors are linearly dependent if $v = aw$ for some vectors $v, w$ and a scalar $a$. It probably has something to do with that $A$ is a linear map, but I don't see how exactly.

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He may mean that $D_vf$ is linear in $v$, so that $D_{v+u}f = D_vf + D_uf$ and $D_{cv}f=cD_vf$ (which are both true, by the way). In other words, $D_vf$ "depends linearly" on $v$, which isn't the same thing as being linearly dependent in the sense of linear algebra/vector space theory.

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