Doubt from Spivak's proof of inverse function theorem.

Question

why must $y^i-f^i(x)=0$ for all $i$?

Answer 1

Note that the equation above can be represented as $$ \begin{bmatrix} D_jf^i(x) \end{bmatrix} ^T \begin{bmatrix} 2(y^1-f^1(x))\\ \vdots \\ 2(y^n-f^n(x)) \end{bmatrix} = \vec0. $$ If matrix $[D_jf^i(x)]^T$ have non-zero determinant (i.e. the inverse exists) then the system must have trivial solution since one can multiply equation above by its inverse. So we have $y^i-f^i(x) = 0$ for all $i=1,\dots,n$.

Answer 2

The equation can be rewritten as a matrix equation: Denote $D_jf^i=M_j^i$ and $2(y^i-f^i)=v^i$, then we have $M_j^i v^i=0$, or $M \vec{v}=\vec{0}$. In other words, it is of the form $(\text{matrix})\cdot (\text{vector})=(\text{zero vector})$.

The assertion that $D_j f^i$ has nonzero determinant implies that $M$ is invertible, hence the equation can only be satisfied if $\vec{v}$ itself is the zero vector, i.e. all components vanish, i.e. $v^i=0$ for all $i$.