So this question is quoted as 'another version of the Implicit Function Theorem, in which we do not specify ahead of time which variables will be implicitly defined ones'. It is also the proof of the 'if' part of a theorem (Manifolds as level sets): $M \subseteq \mathbb{R}^{M}$ is a $C^k$ m-dimensional manifold if and only if it is locally the pre-image of a regular value of a $C^k$ $\mathbb{R}^{M-m}$-valued function. It's set as an exercise in my lecture notes but because of its importance (to my module) I would really appreciate some feedback.
Let $U \subseteq \mathbb{K}^n$ be open, and suppose that $g \in C^1(U;\mathbb{K}^m)$ is such that at $x^* \in U, $dg($x^*$) $\in \mathcal{L}(\mathbb{K}^n;\mathbb{K}^m)$ is surjective.
(a) QUESTION: Show that there exists indices $i_1,....,i_m \in$ {$1,....m$} such that (Sorry, but I don't know how to insert a matrix) the (square) matrix with entries $(\frac{\partial g_a}{\partial x_{i_b}})$ $1 \leq a,b \leq m$ is non singular at $x=x^*$, and indeed for all $x$ in some neighbourhood of $x^*$.
ANSWER: So I basically said since $dg(x^*)$ is surjective, it's rank is $m$ and so there are $m$ linearly independent columns in the corresponding Jacobian Jg$(x^*)$ and so we can use this sub matrix (which is square) and has full rank hence det is non-zero. The result then follows since the det function is continuous.
(b) Show that there is a choice of coordinates $x_1,...,x_n$ on $\mathbb{K}^n$ such that $x_1,...,x_m$ can be written as a well-defined $C^1$ function of $x_{m+1},...,x_{n}$ near $x^*$, and such that the equation $g(x_1,...,x_n) \equiv g(x^*)$ holds.
^ This is the part I'm stuck on, I don't know where to start, I can't do it without using IFT but obviously they don't want us to use that since the question would be pointless.