# Orthogonal Complements of null space and row space

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.....".

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 $$N(A)^\perp = C(A^T)$$.