Statement: I am going to show that if V is a subspace of Rn. then dim(V)+dim(Vorth)=n
Because of the lack of the orthagonal symbol in the wikipedia formatting page, I will be type 'orth' in a superscript to symbolize that.
Analysis:
First, let us say we have the following:
V which is a subspace of Rn, and {v1,v2,v3,..,vk} are a basis for V. (The entries in the braces are vectors)
To refresh, a basis means those entries span V, AND are also linearly independent.
So, therefore, then dim(V)=k (k is the number of vectors in our basis, which obviously is a non-finite amount, so I use k to denote that fact.)
Now that we have those assumptions and definitions out of the way, let me construct a matrix for you.
We will call this matrix A (seems to the most common letter in the linear algebra world...but i digress)
A=
