When reading about orthogonal complement, I encounter the following claim:
If the subspace is described as the range of a matrix: $S = \{ Ax : x \in \Bbb R^n \}$, then the orthogonal complement is the set of vectors orthogonal to the rows of A, which is the nullspace of $A^T$.
I don't quite understand how we can make the above claim from the definition of orthogonal complement as the set of vectors that are orthogonal to all $Ax$. Could someone please explain?
Let $C_1,\ldots,C_p$ be the columns of the matrix $A$, then $S=\operatorname{span}(C_1,\ldots,C_p)$, hence $x\in S^{\perp}$ iff $\langle C_i,x\rangle =0,\;\forall i\in \{1,\ldots,p\}$ iff $A^{T}x=0$. (the lines of $A^{T}$ are the columns of $A$).