dimension of quotient space I am confused about the following:


*

*In Wiki:


 
=> dim(vector space) - dim(subspace) = dim(quotient space)


*In S. Boyd's textbook of cvx (p.22)



=> dim(subspace) = dim(affine set)

Problem:
As far as I know, affine is other name of quotient space (or linear variety).
However, the definition of dimension is different.
Any mistake in my thinking?  (hope no obvious error)
 A: The word affine probably has a dozen meanings, but quotient space is not one of them. 
Quotient space $V/W$ of two vector spaces $V,W,W\subsetneq V$ is itself a vector space, but it is not a subspace of $V$, because elements $v+W$ of $V/W$ are actually affine subsets of $V$ instead of vectors. These affine sets all have the same dimension as $W$. But 
$$\dim V/W=\dim V-\dim W$$ 
as you said. 
To understand the concept of quotient space, consider the scenario in which the information in $W$ is not important to you, and you want to lose it. One can find a complementary subspace $U$ of $W$ in $V$ (assuming that $V$ is finite dimensional) such that:
$$
V=W\oplus U
$$
Any vector $v\in V$ can be uniquely decomposed into components $v_W\in W$ and $v_U\in U$. So you simply dump $v_W$ and keep $v_U$. But the problem is that one usually does not have a natural choice of the complementary subspace $U$. The concept of quotient space is to create instead a unique model for all possible choices of complementary subspaces. More specifically, for the quotient space $V/W$, a natural projection is introduced:
$$
\pi:V\to V/W, \pi(v)=v+W
$$
which is a linear surjection. $\pi$ restricted to any complementary subspace $U$ of $W$ is an isomorphism:
$$
\pi|_U:U\xrightarrow[]{\sim}V/W
$$
