Let $A\in M_{m\times n}(\mathbb R)$. Describe the set of all vectors in $F^{m}$ orthogonal to $Im A$.

Question

Let $A$ be a real $m\times n$ matrix. Describe the set of all vectors in $\mathbb F^{m}$ orthogonal to $Im A$.

source: Linear Algebra Done Wrong by Gilbert Strang.

I'm having trouble understanding this question. Does a real $m\times n$ matrix imply it's going from $\mathbb R^{n}$ to $\mathbb R^{m}$ or it just has real entries? How would we find the set of orthogonal vectors to $Im A$.

Answer 1

Consider the system $$A^Tx = 0$$

The solution describes the set of $x \in \mathbb{R}^m$ that are orthogonal to the columns of $A$.

The nullspace of $A^T$ is orthogonal to the $Ran A$.