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:
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.