Prove or disprove
If $AXX'=0$ then $AX=0$, where $A$ is a square matrix and $X$ is a rectangular matrix.
Update: $X'$ is the transpose of $X$ and the field of this vector space is the set of real numbers.
2026-03-29 20:27:23.1774816043
$AXX'=0$ vs $AX=0$
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Hint. $AXX^T=0$ implies that $\left[(AX)^T\right]^T(AX)^T=AXX^TA^T=0$. If $Y$ is a real (rectangular) matrix such that $Y^TY=0$, can you conclude that $Y=0$? What are the diagonal elements of $Y^TY$?