In Stephen Boyd's optimization book, it says
"We define the dimension of an affine set C as the dimension of the subspace V = C−x0,where x0 is any element of C"
I have two questions regarding this:
Why do we need to introduce the subspace to define the dimension of the affine set ?
Based on my understanding, I am guessing the subspace is a subset which is a vector space and is closed under linear combinations [link]
So the subspace is also an affine set. This leads to a circular argument for defining the dimension of an affine set. I would appreciate some clarity regarding the above points.
The dimension of a linear subspace is well-defined (refer to any linear algebra text); as you noted it is a subset of a vector space that is itself a vector space (closed under linear combinations).
Boyd is defining the dimension of an affine set by producing a related set $C-x_0$ that happens to be a linear subspace, and then referring to the established definition of dimension for linear subspaces. There is no circular argument here.
The set $C-\{x_0\}$ is not a subset of $C$; geometrically, it is shifting/translating the affine space until it touches the origin.