on the rank condition of regression model with cross validation Consider a standard linear regression model in a matrix form given by
\begin{equation}
y=X\beta+u
\end{equation}
where $y$ is $n \times 1$, $X$ is $n \times k$ with $n>k$. Assume that $X$ has full column rank.
When using cross validation, we typically remove some rows of $X$. Let $X_{-i}$ be an $(n-1) \times k$ matrix where $i$th row is removed from $X$.
Then, even when $X$ has full column rank, $X_{-i}$ might have a reduced rank.
I'm wondering under what conditions, $X_{-i}$ is still of full column rank when $X$ is of full column rank.
Or more generally, let us consider a matrix
\begin{equation}
A=\begin{bmatrix}
A_1 \\
A_2
\end{bmatrix}
\end{equation}
where $A_1$ and $A_2$ have more rows than the columns.
I'm wondering under what conditions $A_1$ becomes full rank when $A$ is of full rank.

One condition could be: if all the rows of $A_2$ are in the span of the rows of $A_1$, then $A_1$ is of the same rank as $A$. But that is kind of the definition of rank.
