Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 0 down vote accepted

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''.

share|cite|improve this answer

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.