Consider the hyperplane $$ H := \left\{\mathrm x \in X: \langle \mathrm{w, x}\rangle = b \right\}, $$ and suppose that $\mathrm{e_1} \dots, \mathrm{e_n}$ is a basis for $X$. $H$ isn't necessarily a subspace of $X$ - it need not contain the zero vector - but if one looks up a "Hyperplane" on Wikipedia it is stated that a hyperplane is an affine subspace of $X$. Is it possible to explicitly derive a basis of $n-1$ vectors for this affine subspace $H$; thereby showing that the dimension of $H$ is at most $n-1$?
Thanks!