by Qiang Gao, updated at Mar 21, 2017
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Hint: Use the identity $$ \sum_i (y_i - \bar{y})^2 = \sum_i y_i^2 - n \cdot \bar{y}^2 $$.
By definition of $$ R^2 $$, the left side equals
By definition of $$ R_{uc}^2 $$, the right side equals
$$ \begin{align} & \left( 1 + \frac{ n \cdot \bar{y}^2 } { \sum_{i=1}^n (y_i - \bar{y})^2 } \right) \frac{ \sum_{i=1}^n e_i^2 }{ \sum_{i=1}^n y_i^2 } \ = & \left( \frac{ \sum_i y_i^2 - n \cdot \bar{y}^2
- n \cdot \bar{y}^2 } { \sum_{i=1}^n (y_i - \bar{y})^2 } \right) \frac{ \sum_{i=1}^n e_i^2 }{ \sum_{i=1}^n y_i^2 } \qquad \text{(hint)} \ = & \left( \frac{ \sum_i y_i^2 } { \sum_{i=1}^n (y_i - \bar{y})^2 } \right) \frac{ \sum_{i=1}^n e_i^2 }{ \sum_{i=1}^n y_i^2 } \ = & \frac{ \sum_{i=1}^n e_i^2 }{ \sum_{i=1}^n (y_i - \bar{y})^2 }. \end{align} $$
Left side and right side are equal.
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