The correlation remains the same.
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The correlation coefficient, plus graphical methods to verify the validity of a linear relationship (which is what the correlation coefficient measures), and the appropriate tests of the statisitical significance of the correlation coefficient.
No, it is a linear transformation.
If variables have zero correlation, they do not have a linear relationship. Zero correlation shows that two things were not found to be related.
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No. If the correlation coefficient is close to 1 or -1, then the two variables have a high degree of statistical linear correlation. See the related link, particularly the graphs which illustrate correlation.