John Lee's Intro to Smooth Manifolds Inverse Function Theorem In John Lee's "Intro to Smooth Manifolds" Chapter 7, p 160, we have a proof of the inverse function theorem.
Here, in the middle of the page we have $F_2 = DF(0)^{-1} \circ F$.  Is $DF(0)^{-1}$ a matrix with numbers?  If so, then how can you compose a matrix with numbers with another function $F$?  Is this just multiplying 2 matrices ?
Is $F_2(0)=0$ because $F_1(0) =0$ so you are composing (multiplying) with 0?  
 A: The total derivative of $F$ at $0$ is denoted $DF(0)$. This is based on page 642 where the total derivative is defined. We can either think of this as a matrix, or, as a linear transformation. If $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ then $DF(0)(v) \in \mathbb{R}^n$ for each $v \in \mathbb{R}^n$. Likewise, the inverse $(DF(0))^{-1}$ can either be thought of as a matrix or a transformation. Clearly, if we write $DF(0)^{-1} \circ F$ then we are thinking of $DF(0)^{-1}$ as a linear transformation. That is, $F_2$ is formed by the composition of the function $DF(0)^{-1}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ and $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$.
All of this said, the formula for the function $F_2$ at $p$ amounts to multiplication of the vector $F_2(p)$ by the matrix $DF(0)^{-1}$. If this overloading of notation is too bothersome, you could use $[DF(0)^{-1}]$ for the matrix and $DF(0)^{-1}$ for the transformation. Lee is much clearer than many texts, but, we do assume some linear algebraic insight in pretty much everything we write in manifolds.
If you read the other sentences in the appendix where this proof is found on pages 657-660.
