Chapter 2.4, Problem 48E

Consider an invertible linear transformation $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$ from ${ℝ}^m$ to ${ℝ}^n$ , with inverse $L=T^{-1}$ from ${ℝ}^n$ to ${ℝ}^m$ . In Exercise 2.2.29 we show that L is a linear transformation, so that $L\left(\stackrel{\to }{y}\right)=B\stackrel{\to }{y}$ for some $m\times n$ matrixB. Use the equations $BA=I_n$ , and $AB=I_m$ , to showthat $n=m$ . hint: Think about the number of solutionsof the linear systems $A\stackrel{\to }{x}=\stackrel{\to }{0}$ and $B\stackrel{\to }{y}=\stackrel{\to }{0}$ .