dimension of quotient space

Question

I am confused about the following:

1. In Wiki:

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

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

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

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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)

Answer 1

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