Chapter 6.2, Problem 57E

Consider a linear transformation T from ${ℝ}^{m+n}$ to ${ℝ}^m$ .The matrix A of T can be written in block form as $A=\left[\begin{array}{cc}{A}_1& A_2\end{array}\right]$ . where $A_1$ is $m\times m$ and $A_2$ is $m\times n$ .Suppose that $\det \left(A_1\right)\ne 0$ . Show that for every vectorin R” there exists a unique $\stackrel{\to }{y}$ in ${ℝ}^m$ such that $T\left[\begin{array}{l}\stackrel{\to }{x}\\ \stackrel{\to }{y}\end{array}\right]$ .
Show that the transformation $\stackrel{\to }{x}\to \stackrel{\to }{y}$ from ${ℝ}^n$ to ${ℝ}^m$ is linear, and find its matrix M (in termsof $A_1$ and $A_2$ ). (This is the linear version of the implicit function theorem of multivariable calculus.)