# Proof that if covariance is zero then there is no linear relationship

I get that a zero covariance doesn´t imply independence, but everybody says that if there is dependence and the covariance is zero then it is a non linear dependence.

People base their interpretation of Pearson’s R in that fact (the closer you are to zero the less linear the relationship is).

Is there a formal proof to that?

I tried to do it by myself but i couldn’t. The proposition i think encapsulates the idea is the following:

If $$cov(X,Y)\ne0$$ then there exists a Z such that $$cov(X,Z)=0$$ and $$E[Y|X]=bX+E[Z|X]$$

Here is a proof of the mathematical statement at the end of your question: we can find a $$Z$$ which is uncorrelated to $$X$$ and satisfies
$$\mathbb{E}(Y|X) = b X + \mathbb{E}(Z|X)$$
by assuming $$Z = Y – bX$$, and then choosing the $$b$$ which makes $$\mathrm{Cov}(X, Z) = 0$$ true. For this $$b$$ we have
$$0 = \mathrm{Cov}(X, Z) = \mathrm{Cov}(X, Y – bX) = \mathrm{Cov}(X, Y) – b \mathrm{Var}(X),$$
$$b = \frac{\mathrm{Cov}(X, Y)}{\mathrm{Var}(X)}.$$
(Note that the same $$b$$ is found as the slope of the linear regression line.) We have $$b = 0$$, if and only if $$\mathrm{Cov}(X,Y) = 0$$.