# 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.....".

## 1 Answer

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