Commensurability for vector spaces Let me start by saying I am a not a mathematician and I have not studied group theory (just a few brushes here and there) but after reading I have a very basic understanding of commensurability as defined here. Intuitively (and quite crudely/non-rigorously) I understand it thus: 
First, every subgroup of a group induces an equivalence class on the group (I am not concerned with how for the purposes of this question). Second, the number of equivalence classes a subgroup induces on the group is called the index; denoted as  $ [G_0 : G ] $ where $G_0$ is a subgroup of $G$. Now given two subgroups $ G_1 $ and $ G_2$ of $ G $ define the quantity $[ G_j : G_1 \cap G_2 ]$ where $j=1,2$. If this quantity is finite then $ G_1 $ and $G_2$ are said to commensurable. 
(Please feel free to correct me if the above is wrong; but try and not chide me for lack of formalism)
Now, I want help to understand this in the vector space case. Wikipedia says, 

A relationship can similarly be defined on subspaces of a vector space, in terms of projections that have finite-dimensional kernel and co-kernel.

So I am going to assume that similar to the above what this means is given some vector space $V$ and sub spaces $V_1$ and $V_2$; they are commensurate if the some projection(s) involving them has the basis for the kernel and co-kernal finite dimensional. But what exactly are the projections? What are we projecting onto where?
Thank you for the help. 
 A: This book provides the definition:

Let $k$ be an arbitrary field, let $V$ be a $k$-vector space and let $A$, $B$ be two $k$-vector subspaces of $V$. $A$ and $B$ are said to be commensurable if $A+B\,/\,A\cap B$ is a finite-dimensional vector space over $k$.

No projections are mentioned, and rightly so: A projection of a vector space onto a subspace introduces additional structure which is not necessary to define the quotient or the index, just as it is not necessary to introduce a homomorphism of a group onto a subgroup to define the quotient or the index.
For example, over the field $k=\mathbb R$, let $A$ be the vector space of sequences of real numbers in which only even-numbered terms are non-zero, except the first term may also be non-zero, and let $B$ be the vector space of sequences of real numbers in which only even-numbered terms are non-zero, except the third term may also be non-zero. Then $A+B$ is the vector space of sequences of real numbers in which even-numbered terms and the first and third terms may be non-zero, $A\cap B$ is the vector space of sequences of real numbers in which only even-numbered terms are non-zero, and $A+B\,/\,A\cap B$ is the vector space of equivalence classes characterised by the first and third terms common to all their elements. This quotient has finite dimension $2$, so $A$ and $B$ are commensurable. The index of $A\cap B$ in both $A$ and $B$ is $1$.
We may project $A$ onto $A\cap B$ e.g. by setting the first term to zero, but this is only one possible projection; another is to subtract the first term from both the first and the second term.
