Clarification - Support Vectors decrease no. of features of W

[a.sanyal902]

In Lecture 14, Professor mentions that because only the support vectors count towards W (the rest have alpha=0) which leads to a decrease in the number of features and thus, better generalization.

I'm not sure I got this point because I thought the VC dimension for W would be equal to d, the no. of dimensions of the space, regardless of the number of points being summed. Aren't we just summing the various "x" vectors, multiplied by alpha*y ? How does this decrease the number of features of W?

Thank You!