A previous question received a linked answer as below:

The first important distinction here is that linear independence and orthogonality are properties of the raw variables, while zero correlation is a property of the centered variables


The key to appreciating this distinction is recognizing that centering each variable can and often will change the angle between the two vectors.

I'm having a hard time visualizing what "centering a variable" means and how it may change the angle. Could anyone produce an image?

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    $\begingroup$ I think the distinction is pretty clear. Orthogonality means $X'X=0$, while uncorrelated means $(X-\bar{X})'(X-\bar{X})=0$. $\endgroup$ – Heisenberg Jan 29 '14 at 4:44
  • $\begingroup$ Ah, I think I misunderstood where your difficulty lies. $\endgroup$ – Glen_b Jan 29 '14 at 4:49
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    $\begingroup$ +1, this is a good question. Unfortunately, I think the necessary picture would need to be in a high-dimensional space. In other words, I'm not sure how well it can be pictured. $\endgroup$ – gung Jan 29 '14 at 5:25
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    $\begingroup$ If it's just 2 variables $X$ and $Y$, shouldn't the picture simply be in $R^2$? $\endgroup$ – Heisenberg Jan 29 '14 at 5:28
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    $\begingroup$ I may recommend you visualization in the subject space. Pearson r between two vectors is the cosine between them when the vectors are centered. Centering is bringing the mean to 0. Centered vector (variable) X is $X_i-\mu$. $\endgroup$ – ttnphns Jan 29 '14 at 6:19

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