I'm reading the book "Foundations of Data Science" by Blum, Hopcroft. In chapter 4, "Random walks and Markov chains" on page 80, it is trying to prove the "fundamental theorem of Markov chains." Beforehand, it mentions lemma 4.1:
Let $P$ be the transition probability matrix for a connected Markov chain. The $n \times (n+1)$ matrix $A = [P-I ; 1]$ obtained by augmenting the matrix $P-I$ with an additional column of ones has rank $n$.
The proof is started with, if assuming $A$ has a rank lower than $n$, then the null space of $A$ must be at least two-dimensional subspace. As we know that one answer to $A\textbf{x}=0$ is $\textbf{x}=(\textbf{1};0)$ which all elements are one except the last zero element. Hence we must prove that there is no answer to $A\textbf{x}=0$ that is perpendicular to $\textbf{x}$.
So it assumes that there is such perpendicular answer $(\textbf{x},\alpha)$ and tries to find a contradiction on the value of $\alpha$.
This is where I am stuck and baffled. The proof seems to sort the $x_i$s based on their value and said that some of them are more than zero, and some are less than zero. And then mentions just one sentence about them comprising a convex combination set. Then surprisingly and without any other explanation, it concludes that $\alpha>0$. Using a similar discussion concludes that for the minimum $x_i$, set the $\alpha<0$. And therefore, this contradicts. So there is no solution in this form, and hence the null space is only one-dimensional!
The book really sucks at explanation.
I appreciate it if you can make an explanation for the last part. Or mention a reference that has the proof of this lemma in a more human fashion.