Q: Set of matrices differentiable manifold? Let $X$ be a set of matrices $2\times 3$, that for all $A$ from $X$ rank $A=1$. Is $X$ a manifold? If not find a maximum subset in $X$, which is a manifold and its dimension. 

A: Let $U_1$ be the set of elements of $X$ whose first row is nonzero, and let $U_2$ be the set of elements of $X$ whose second row is nonzero.  Note that $U_1$ and $U_2$ are open subsets of $X$ whose union is $X$.
Now, every element of $U_1$ can be written uniquely as
$$
\begin{bmatrix}
v_1 & v_2 & v_3 \\
tv_1 & tv_2 & tv_3
\end{bmatrix}
$$
where $(v_1,v_2,v_3)$ is a nonzero vector in $\mathbb{R}^3$, and $t\in\mathbb{R}$.  Thus $U_1$ is $4$-manifold isomorphic to $(\mathbb{R}^3-\{\textbf{0}\})\times \mathbb{R}$.  Similarly, every element of $U_2$ can be written uniquely as 
$$
\begin{bmatrix}
tv_1 & tv_2 & tv_3 \\
v_1 & v_2 & v_3
\end{bmatrix}
$$
So $U_2$ is also a $4$-manifold.  It follows that $X$ is a $4$-manifold.
