Orthogonal Complements of null space and row space

Question

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.....".

Answer 1

From the second paragraph (the paragraph after the definition), we know that all elements of the column space are orthogonal to the nullspace. That is, we can deduce that $C(A^T) \subseteq N(A)^\perp$.

From the third paragraph, we know that every $v$ that is orthogonal to the nullspace is also an element of the row space. That is, $N(A)^\perp \subseteq C(A^T)$.

Because $N(A)^\perp \supseteq C(A^T)$ and $N(A)^\perp \subseteq C(A^T)$, it must be the case that $N(A)^\perp = C(A^T)$.