The quadratic kernel provides a clear example of an implicit feature mapping. Instead of explicitly computing feature mappings $f(\vec{x})$ and $f(\vec{z})$ and then calculating their dot product $f(\vec{x}) \cdot f(\vec{z})$, the kernel efficiently computes this inner product in one step.

To see how the kernel and an explicit feature mapping relate, consider input vectors in $\mathbb{R}^2$. The explicit mapping to a higher-dimensional space, $f([x_1, x_2]^T) = [x_1^2, x_2^2, \sqrt{2}x_1x_2, \sqrt{2}x_1, \sqrt{2}x_2, 1]^T$, mathematically corresponds directly to the quadratic kernel function:

$K(\vec{x},\vec{z}) = (\vec{x} \cdot \vec{z} + 1)^2$

The accompanying proof visually demonstrates that evaluating $K(\vec{x},\vec{z})$ achieves the same exact result as the dot product of the transformed features, allowing the model to avoid the computationally expensive mapping step.