While doing googling , I encountered next AI's answer :
"Elementary row and column operations on the Jacobian matrix of a smooth map between open subsets of Euclidean space correspond to reordering coordinates in the domain or codomain, and these operations affect the Jacobian in predicctable ways."
Here's a more detailed explanation :
Elementary Row Operations and (Co-)Domain Coordinate Changes :
Applying an elementary row operation ( swapping rows, multiplying a row by a scalar, or adding a multiple of one row to another ) to the Jacobian matrix of a smoooth map $f: U \subset \mathbb{R}^n \to \mathbb{R}^m$ is equivalent to changing the order of the output coordinates ( i.e., the coordinates in $\mathbb{R}^{m}$ ). Specifically:
- Swapping rows corresponds to swapping the corresponding output coordinates.
- Multiplying a row by a scalar corresponds to multiplying the corresponding output coordinate by that scalar( after the change of coordinates ). ( What does this mean? )
- Adding a multiple of one row to another corresponds to a similar linear transformation of the corresponding ouput coordinates. ( What does this mean? )
Elementary Column Operations and Domain Coordinate Changes :
Similary, applying an elementary column operation to the Jacobian matrix corresponds to changing the order of the input coordinates ( i.e., the coordinates in $\mathbb{R}^{n}$ ).
- Swapping columns corresponds to swapping the corresponding input coordinates.
- Multiplying a column by a scalar corresponds to multiplying the corresponding input coordinate by that scalar( after the change of coordinates ).
- Adding a multiple of one column to another corresponds to a similar linear transformation of the corresponding input coordinates.
Q. Can anyone explain this AI's answer more friendly, possibly with example?
This quesiton originates from follwing argument ( C.f. Refer to first question in Proof of the Rank Theorem, Lee's smooth manifold book. that I posted ) : Let $F : U \to V$ be a smooth map with $U \subseteq \mathbb{R}^{m}$ and $V \subseteq \mathbb{R}^{n}$. Let $ p \in M$. Assume that the total derivative at $p$, $DF(p)$, has rank $r$. Then it implies that its matrix has some $r \times r$ submatirx with nonzero determinant. By reordering the coordinates, we may assume that it is the upper left submatrix. We want to understand this bold statement. Through elementary operations, the $r \times r$ submatrix can be transformed to the upper left submatrix of $DF(p)$. And using the correspondence between the elementary operations on the Jacobian matrix and the reordering coordinates above, I think we can check the bold statement somehow. Can anyone help?
