I am reading a section of linear algebra book. And I am not sure whether my understanding is correct or not for the sentence that I highlight.
DEFINITION: The orthogonal complement of a subspace $V$ contains every vector that is perpendicular to $V$. This orthogonal subspace is denoted by $V^\perp$. (pronounced "V perp").
By this definition, the nullspace is the orthogonal complement of the row space. Every $x$ that is perpendicular to the rows satisfies $Ax = 0$, and lies in the nullspace.
The reverse is also true. If $v$ is orthogonal to the nullspace, it must be in the row space. Otherwise we could add this $v$ as an extra row of the matrix, without changing its nullspace. The row space would grow, which breaks the law $r + ( n - r) = n$.
We conclude that the nullspace complement $N(A)$ orthogonal is exactly the row space $C(A^T)$.
I understand every sentences separated. I see adding $v$ that breaks the law. I don't understand why suddenly jump to "We conclude.....".