Vector Space Codimension
In the context of a finite-dimensional vector space V over a field K, and a subspace W of V, the codimension is the difference between the dimensions of the space V and the subspace W.
$$ codim_k(W) = dim_k(V) - dim_k(W) $$
The codimension is denoted by the notation codim or codimk.
If the field is clear from context, the specific mention of field k can be omitted.
Note. The dimension of a vector space or a subspace is always a non-negative integer. Thus, the codimension is also a non-negative integer.
What is the Purpose of Codimension?
Codimension is a ratio between the vector space and its subspace.
It measures how many times smaller the subspace W is compared to the vector space V that contains it.
