TITLE: The orthogonal complement of the null space of $A$ is equal to the range of the transpose of $A$
QUESTION [1 upvotes]: Prove $N(A)^{OC} = R(A^T)$, where $()^{OC}$ means orthogonal complement and $A$ is a matrix.

I figured out $R(A^T) \subseteq N(A)^{OC}$ but I'm stuck on the other direction.
I see that if $v \in N(A)^{OC}$ then $(\forall w \text{ such that } Aw=0)(w^Tv = 0)$, but I'm having trouble showing that $v = A^Tx$ for some $x$.  I thought I could combine $Aw = 0$ and $w^Tv = 0$ but one $0$ is a vector and the other is a scalar so I'm not sure how to go about this.
Anyone have any ideas?

REPLY [0 votes]: Sure.  Because as is well known,  and fairly easy,  the null space is the orthogonal complement of the row space.  Put this together with the fact that the image is the column space (and the rows are the columns of the transpose).
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