Column Space after Left-Multiplication by an Invertible Matrix

Question

Suppose that $A$ is an $n\times n$ positive definite matrix and that $B$ is a full-rank $n\times p$ matrix, with $p<n$. Is it the case that the column space of $A^{-1}B$ coincides with the column space of $B$, that is, $\mathcal{C}(A^{-1}B)=\mathcal{C}(B)$?

Answer 1

I think I may have found a counter example. Let $A$ be the following $3 \times 3$ matrix:

$$\left[\begin{matrix}2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2\end{matrix}\right]$$

We now have:

$$\left[\begin{matrix}\frac 3 4 & -\frac 1 2 & \frac 1 4 \\ -\frac 1 2 & 1 & -\frac 1 2 \\ \frac 1 4 & -\frac 1 2 & \frac 3 4\end{matrix}\right]$$

Then, let $B$ be the following $3 \times 2$ matrix:

$$\left[\begin{matrix}1 & 0 \\ 0 & 1 \\ 0 & 0\end{matrix}\right]$$

Then, the column space of $B$ is the plane $z=0$, but the column space of $A^{-1}B$ clearly has vectors outside of this plane since the first column of $A^{-1}B$ has a non-zero z-coordinate, so the column spaces are not the same.