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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

and

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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