I am reading "Ordinary Differential Equations" by L. S. Pontryagin (translated from Russian to Japanese).
Suppose $u^1(x^1,\dots,x^n),u^2(x^1,\dots,x^n),\dots,u^k(x^1,\dots,x^n)$ are functions of class $C^1$ defined on a neighborhood of a point $(a^1,\dots,a^n)$.
If $$\begin{pmatrix}
\frac{\partial u^1(\overrightarrow{a})}{\partial x^1} & \dots & \frac{\partial u^1(\overrightarrow{a})}{\partial x^n}\\
\frac{\partial u^k(\overrightarrow{a})}{\partial x^1} & \dots & \frac{\partial u^k(\overrightarrow{a})}{\partial x^n}\\
\end{pmatrix}$$ has rank $k$, then the following matrix-valued function defined on a neighborhood of $(a^1,\dots,a^n)$ $$\begin{pmatrix}
\frac{\partial u^1(\overrightarrow{x})}{\partial x^1} & \dots & \frac{\partial u^1(\overrightarrow{x})}{\partial x^n}\\
\frac{\partial u^k(\overrightarrow{x})}{\partial x^1} & \dots & \frac{\partial u^k(\overrightarrow{x})}{\partial x^n}\\
\end{pmatrix}$$ has rank $k$ on some neighborhood of $(a^1,\dots,a^n)$ because each $\frac{\partial u^i(\overrightarrow{x})}{\partial x^j}$ is a continuous function defined on a neighborhood of a point $(a^1,\dots,a^n)$.
In this case we say that the functions $u^1,\dots,u^k$ are independent.
If $$\begin{pmatrix} \frac{\partial u^1(\overrightarrow{x})}{\partial x^1} & \dots & \frac{\partial u^1(\overrightarrow{x})}{\partial x^n}\\ \frac{\partial u^k(\overrightarrow{x})}{\partial x^1} & \dots & \frac{\partial u^k(\overrightarrow{x})}{\partial x^n}\\ \end{pmatrix}$$ has rank less than $k$ on some neighborhood of $(a^1,\dots,a^n)$, then we say that the functions $u^1,\dots,u^k$ are dependent.
My question is the following:
If $$\begin{pmatrix} \frac{\partial u^1(\overrightarrow{a})}{\partial x^1} & \dots & \frac{\partial u^1(\overrightarrow{a})}{\partial x^n}\\ \frac{\partial u^k(\overrightarrow{a})}{\partial x^1} & \dots & \frac{\partial u^k(\overrightarrow{a})}{\partial x^n}\\ \end{pmatrix}$$ has rank $k'$ which is less than $k$, then can we consider the rank of the following matrix-valued function defined on a neighborhood of $(a^1,\dots,a^n)$ $$\begin{pmatrix} \frac{\partial u^1(\overrightarrow{x})}{\partial x^1} & \dots & \frac{\partial u^1(\overrightarrow{x})}{\partial x^n}\\ \frac{\partial u^k(\overrightarrow{x})}{\partial x^1} & \dots & \frac{\partial u^k(\overrightarrow{x})}{\partial x^n}\\ \end{pmatrix}$$ on some neighborhood of $(a^1,\dots,a^n)$?
I know that the rank of the value of the above matrix-valued function is at least $k'$ for each $\overrightarrow{x}$ on a neigborhood of $\overrightarrow{a}$.