Why does an SVM model store the support vectors, and not just the separating hyperplane?
The hyperplane is a linear combination of the support vectors. In the soft margin case, there is only a limited amount of slack; every input does not get to be support vector. In the nonlinear case, the separating hypersurface may be embedded in an infinite-dimensional space, making it impossible to store. To borrow from the Wikipedia article, the normal vector $w$ is given by

$$w = \sum_i c_i y_i \phi(x_i)$$

where $\phi$ is the feature embedding function, and $c_i$ is a Lagrangian dual variable that is zero for points on the correct side of the margin. Instead, test points are classified through a kernel function $k(x_i,x_j) = \left&lt; \phi(x_i), \phi(x_j) \right&gt;$ like so:

$$x \to \mathrm{sgn}(\left&lt;w , \phi(x)\right&gt; + b) \equiv \mathrm{sgn} \left( b+\sum_i c_i y_i k(x_i, x)\right)$$

Notice how we avoided explicitly calculating $w$.