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)
Best Answer
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: $$ \pi|_U:U\xrightarrow[]{\sim}V/W $$