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Given a set $\{f_{1},\ldots,f_{k}\}$ of maps $\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ how is it usually defined the notion of linear (in)dependence among them? (Is it the same as for single variable real valued functions?).

Does the notion of linear (in)dependence have an analog, or a generalization to maps between more general types of spaces? (e.g. maps between Banach or Hilbert spaces, or even metric spaces)

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    @John You should not confuse the definition of linear dependence with the tricks you can do in single, special cases. For functions, the definition of linear dependence is a pointwise one: $\sum_{i=1}^n \alpha_i f_i(x)=0$ at every point $x$. Maybe you can take derivatives to get some additional information, but this depends on the choice of the vector space.2012-09-24

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It's the same. The set is linearly independent if there is no non-zero solution to $ a_1f_1+\cdots a_kf_k=0 $ for $a_1,\ldots ,a_k\in \mathbb{R}$.

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    @Thomas. +1 to your comment. The point you make precisely is where my confusion came from. The way it is written it seems that both parts are part of the same definition when the functions are $C^{n-1}$.2012-09-24