# Does Kernel Function Only Apply To Support Vector In SVM?

We know that the if $$α=0$$ in below equation it is not a support vector; support vectors have $$α \ne 0$$.

$$L(w, b, α) = \sum_{i=1}^m α_i − \frac12 \sum_{i,j=1}^m y^{(i)} y^{(j)} α_i α_j (x^{(i)})^T x^{(j)}.$$

However, the kernel function i calculated for all pairs of vectors even if $$α=0$$.

Can we say that kernel is calculated for all vectors, but only support vector contribute to the final decision boundary?

Or actually is the kernel only calculated for support vectors?

It confuse me that because only support vector decide the boundary, why is the kernel need to calculate all pair of vectors in sample?