My lecture notes leading up to the implicit function theorem state the following:
We investigate underdetermined systems of equations $f(x) = b$ where $f$ : $\mathbb{R}^{n+k} \to \mathbb{R}^k$ and $b \in \mathbb{R}^k$ is given. Note that it is sufficient to consider the case $b = 0$ since we can subtract $b$ from $f$. Thus we consider the solution space $M := f^{-1}(0) \subset \mathbb{R}^{n+k}$.
Each component of $f$ gives rise to an equation. Writing $M = \{x : f_1(x) = 0\} \cap... \cap \{x : f_k(x) = 0\}$ we see that each scalar equation reduces the dimension of the solution space by one. Thus we expect $M$ to be $n$-dimensional with codimension $k$. Our task is to find a way to parameterize the set of solutions.
Now for a subspace of a vector space it is clear what dimension means, but the solution set as a subset of a vector space is not necessarily a vector space. So my question is how to define dimension in this case?
Wikipedia mentions that:
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one (1D) because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces.
Is there a more rigorous definition? Is this why we introduce the concept of manifolds?
Thanks very much!