Statement: I am going to show that if V is a subspace of Rⁿ. then dim(V)+dim(V^orth)=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,..,v_k} 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:

$ \begin{bmatrix} v_{1} & v_{2} & v_{3} & v_{k} \\ v_{1} & v_{2} & v_{3} & v_{k} \\ v_{1} & v_{2} & v_{3} & v_{k} \\ .... & .... & .... & .... \\ .... & .... & .... & .... \\ .... & .... & .... & .... \\ v_{1} & v_{2} & v_{3} & v_{k} \\ \end{bmatrix} $