The Jacobian, change of basis, and row-equivalence

Question

This question concerns the connections between multivariable calculus and linear algebra. In particular, I'd like to verify certain notions about Jacobian matrix.

For the following questions, please assume the following definition for differentiability:

Definition(1): Let $\mathbf{f} : D \rightarrow \mathbb{R}^m$, where $D \subset \mathbb{R}^n$ is open, be a function, and let $\mathbf{a}$ be a point in $D$. Then $\mathbf{f}$ is differentiable at $\mathbf{a}$ if there exists a linear function $L : \mathbb{R}^n \rightarrow \mathbb{R}^m$ (defined by $L : \mathbf{x} \mapsto A\mathbf{x}$) such that $$\lim_{\mathbf{h} \rightarrow \mathbf{0}}{\frac{\|\mathbf{f}(\mathbf{a} + \mathbf{h}) - \mathbf{f}(\mathbf{a}) - A\mathbf{h}\|}{\|\mathbf{h}\|}} = 0 \tag1$$

Further:

• Let $D(\mathbf{a})$ denote the set of all functions differentiable at $\mathbf{a}$.
• Let $\mathcal{M}_{\mathbf{f}}$ denote the set of all matrices row equivalent to the standard Jacobian of a function $\mathbf{f} \in D(\mathbf{a})$.

Let $\mathbf{f} : D \rightarrow \mathbb{R}^m$, where $D \subset \mathbb{R}^n$ is open, be a function given by:

$$\mathbf{f} = \begin{bmatrix} f_1\\ f_2\\ \vdots\\ f_m\\ \end{bmatrix},$$

and suppose $\mathbf{f} \in D(\mathbf{a})$. The standard Jacobian of $\mathbf{f}$ is given by:

$$\mathbf{J}_{\mathbf{f}} = \begin{bmatrix} D_1f_1 & D_2f_1 & \dotsb & D_nf_1\\ D_1f_2 & D_2f_2 & \dotsb & D_nf_2\\ D_1f_1 & \vdots & \ddots & \vdots\\ D_1f_m & D_2f_m & \dotsb & D_nf_m\\ \end{bmatrix}$$

1. Suppose $\mathbf{f} \in D(\mathbf{a})$. For all $A \in \mathcal{M}_{\mathbf{f}}$, does there exist a linear transformation $T : \mathbf{f} \mapsto T(\mathbf{f})$ such that $A$ satisfies (1) for $T(\mathbf{f})$?

My thoughts:

Any row operation on $\mathbf{J}_{\mathbf{f}}$ to produce a matrix $\mathbf{J}_{\mathbf{f}}^*$ can be represented by multiplication by an elementary matrix $E$. This same matrix can be used to define a function $\mathbf{g} = E\mathbf{f}$ such that $\mathbf{J}_{\mathbf{g}} = \mathbf{J}_{\mathbf{f}}^*$. I think that such a $T$ corresponds to the change of base matrix in $\mathbb{R}^m$.

2. Suppose $\mathrm{col}\, B \subset \mathrm{col}\, \mathbf{J}_{\mathbf{f}}$ for some matrix $B$. Does there exist a transformation $S : \mathbf{f} \mapsto S(\mathbf{f})$ such that $B$ satisfies (1) for $S(\mathbf{f})$?