I am trying to tackle question 2.3.8 on GP, but I haven't figure out the following question yet.
Suppose $m > 1$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth map. Consider $f + Ax$ for $A \in \mathrm{Mat}_{m\times n}$. Define $F: \mathbb{R}^n \times \mathrm{Mat}_{m\times n} \rightarrow \mathrm{Mat}_{m\times n}$ by $F(x,A) = df_x + A$. How can I show that $dF_x$ is surjective for all $x$?
I am really grateful to @Ross B. 's answer on the question The derivative of a linear transformation, $DF$ would be a rank-3 tensor with elements
$$ (DF)_{i,j,k} = \frac{\partial^2 f_i}{\partial x_j \partial x_k} $$
Some authors also define matrix-by-vector and matrix-by-matrix derivatives differently be considering $m \times n$ matricies as vectors in $\mathbb{R}^{mn}$ and "stacking" the resulting partial derivatives.
Then I got really lost trying to show that $dF$ is surjective.
Some thoughts I had so far:
$f: \mathbb{R}^n \rightarrow \mathbb{R}^m$, and its tangent plane has dimension $m \times n$, therefore $dF$ maps from $m \times n \times n$ to $m \times n$. Then I am thinking of to prove that $\forall m \in \mathrm{Mat}_{m\times n}, \exists x \in \mathbb{R}^n, A \in \mathrm{Mat}_{m\times n}$ such that $ d f_x + A = m$. I am not sure if this is correct, nor how to prove this.
Thank you very much.
This question arises many very interesting discussions: like The dimension of the derivative and The derivative of a linear transformation. These discussions are really very helpful and inspiring for me. Thanks all the wonderful teachers!
However, the solution to show that $dF_x$ is surjective for all $x$ is rather straight forward, thanks to Professor @Ted Shifirin's patient guidance. I hope I carried out Ted's beautiful proof correctly:
Consider $\displaystyle dF_{x,A}(0,B) = \lim_{t \rightarrow 0}\frac{F(x,A+tB)-F(x,A)}{t}= \lim_{t \rightarrow 0}\frac{df_x + A+ tB-df_x - A}{t}= B$. Therefore, $\forall m \in \mathrm{Mat}_{m\times n}, \exists B \in \mathrm{Mat}_{m\times n}$, such that $dF_{x,A}(0,B) = m$. In particular, $B=m$.