Kernel Function:
\[ x'= \begin{bmatrix} (x_1 - \bar{x}_1)^2 \\ (x_2 - \bar{x}_2)^2 \end{bmatrix} = (x - \bar{x})^{\top} (x - \bar{x}) \]
Step 1. Generate a set of random points. We observe that these points are not linearly separable. To address this, we apply the transformation using the given formula.
Step 2. Select a point \( (\bar{x}_1,\bar{x}_2) \) at random and assume that \( \bar{x}_1 = 0,\bar{x}_2 = 0\).
Step 3. Click the "Transform" button to visualize the transformation process through an animation based on the provided formulas.
Example: An example is provided for reference to illustrate the transformation.
\[ \begin{bmatrix} 1.5 \\ 2 \end{bmatrix} \to \begin{bmatrix} (1.5 - 0)^2 \\ (2 - 0)^2 \end{bmatrix} \to \begin{bmatrix} 2.25 \\ 4 \end{bmatrix} \]
Result: The original point \( (1.5, 2) \) is mapped to \( (2.25, 4) \) after projection.
Conclusion: After applying the transformation, all points that were originally non-linearly separable become linearly separable, demonstrating the effectiveness of the projection.