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I'm not sure what the proper terms are here, so I figure it's better to illustrate with examples.

If I look at the family of polynomials of a certain degree (e.g cubics), the coefficients in front of each term are independent. So a general cubic such as: $$ax^3+bx^2+cx+d, a \neq 0$$ has $4$ degrees of freedom, as it were. However if I were to talk about a family of, say exponentials:

$$a\cdot b^{cx+d}$$

It would turn out that I really only have $2$ 'degrees of freedom', because the $b^d$ can be incorporated into the arbitrary constant $a$, and the $b$ itself can be incorporated into the arbitrary constant $c$ such that the base is fixed (or vice-versa).

Is there a general way of knowing how many of these arbitrary constants are not redundant? I was hoping that I could get some insights from linear algebra in terms of linear independence, but I don't see a general solution to the problem that way.

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Simple answer : no. In general, you have to work ; life is not that easy. –  Patrick Da Silva Jun 8 '12 at 17:45
Relevant: Given $n$ scalar-valued functions on an interval, their Wronskian is a single scalar-valued function that is identically zero if the functions are linearly dependent. –  Rahul Jun 8 '12 at 18:41
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Not sure the best way to express this, but a fairly practical test seems to be: check for linear relations among the partial derivatives. Example: $ax^3+bx^2+cx+d$ has derivatives $x^3$, $x^2$, $x$, 1, with respect to $(a,b,c,d)$, and the derivatives are linearly independent. Example: by inspection there are only two linearly independent derivatives of $f=ab^{cx+d}$. A relation such as $adf_a+\log(b)f_b-cf_c=0$ shows that a small change in $a$, say, holding for all $x$, cannot occur without some change in $b$ or $c$.

Of course there might be more complicated relations than linear ones. I think the classical terminology that you want is ``functional dependence''.

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