Suppose that there is a dataset of 2D points $(x_i, y_i)$, consider the following statement:
"When the Pearson Correlation Coefficient(PCC) between $x$ and $y$ is equal to -1 or 1 (highest absolute values), data could be described by a straight line". That is, there are constants $a$ and $b$ such that:
$y_i = ax_i+b \;\;\: \forall i$
Now, I have two questions:
1) Is the above statement true?
2) More importantly, if the statement is true, please give a mathematical proof.
I have visually seen that a line can be fitted perfectly on such a dataset, but I am not sure this is always true, so I need a mathematical proof.