I've recently learned that the column space of a matrix A that can be decomposed through QR decomposition is the same as the column space of Q (refer to here). Since $\overline A^T=\overline R^T\overline Q^T$, does this mean that the column space of $\overline A^T$ is the same as the column space of $\overline R^T$? Or is it something else?
Note: $\overline A^T$ is the conjugate transpose of A
The column space of $A$ is the same as the column space of $Q$ because the $QR$ decomposition is done by taking the column vectors of $A$, a basis of the column space, and creating an orthonormal basis from it using the Gram-Schmidt procedure. It simply chooses a nicer basis for the column space of $A$, so the column space of A is the same.
The column space of $A^T$ is called the row space of $A$. $A$ and $Q$ having the same column space does not imply they have the same row space.
You can prove that for a product $AB$, the column space of $AB$ is a subset of the column space of $A$, and since $(AB)^T = B^T A^T$, we can say that for a product $AB$, the row space of $AB$ is a subset of the row space of $B$.
More can be said if $A$ or $B$ is invertible.