The problem I am trying to solve is:
Prove that the set of $m \times n$ is matrices of rank $r$ is a submanifold of $\mathbb{R}^{mn}$ of of codimension $(m - r)(n -r)$. [HINT: Suppose, for simplicity, that an $m \times n$ matrix $A$ has the form $$A = \begin{pmatrix} B & C \\ D & E \end{pmatrix},$$ where the $r \times r$ matrix $B$ is nonsingular, $C$ is $r$ row $n-r$ column, $D$ is $m-r$ row, $r$ column, and $E$ is $m-r$ row, $n-r$ column. Postmultiply by the nonsin-gular matrix $$\begin{pmatrix} I & -B^{-1}C\\ 0 & I \end{pmatrix}$$ to prove that rank$(A)= r$ if and only if $E - DB^{-1}C = 0$.]
In order to show $U \cap M$ is a submanifold of $U$ of codimension $(m-r)(n-r)$, it suffices to show that 0 is a regular value of $E - DB^{-1}C$. That is to show that every $(m-r)(n-r)$ matrix $F$ is in the image of the derivative of $E - DB^{-1}C$, But,
But what does is mean by differentiate $E - DB^{-1}C$? Particularly, how to differentiate it in the direction $$\mathfrak{F} = \begin{pmatrix} 0 & 0 \\ 0 & F \end{pmatrix}?$$
I tried according to definition: \begin{eqnarray*} d(E - DB^{-1}C)_\mathfrak{F}(X) & = &\lim_{t\to 0}\frac{(E - DB^{-1}C)(\mathfrak{F} + tX) - (E - DB^{-1}C)(\mathfrak{F})}{t}\\ \text{By linearity:}& = & \lim_{t\to 0}\frac{(E - DB^{-1}C)(\mathfrak{F}) + t(E - DB^{-1}C)(X) - (E - DB^{-1}C)(\mathfrak{F})}{t}\\ & = & (E - DB^{-1}C)(X)\\ \end{eqnarray*}
So I got the derivative of $E - DB^{-1}C$ at the direction of $F$ is $E - DB^{-1}C$, how can I draw the conclusion that it is surjective? As a matter of fact, I don't even sure if it is a good idea to differentiate in the $\mathfrak{F}$ direction, since $E - DB^{-1}C$ is $m-r$ row, $n-r$ column matrix.
This disagrees with other people's work, whose derivative results $F$ rather than identity.
Thanks