Chapter 5.4, Problem 8E

Consider a linear transformation $L\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$ from ${ℝ}^n$ to ${ℝ}^m$ . with $\ker \left(L\right)=\left\{0\right\}$ . Thepseudoinverse $L^{+}$ of Lis the transformation from ${ℝ}^m$ to ${ℝ}^n$ given by
$L^{+}\left(\stackrel{\to }{y}\right)=$ (the least-squares solution of $L\left(\stackrel{\to }{x}\right)=\stackrel{\to }{y}$ ).

a. Show that the transformation $L^{+}$ is linear Find thematrix $A^{+}$ of $L^{+}$ . in terms of the matrix A of L.
b. If L is invertible, what is therelationship between $L^{+}$ and $L^{-1}$ ?
c. What is $L^{+}\left(L\left(\stackrel{\to }{x}\right)\right)$ , for $\stackrel{\to }{x}$ in ${ℝ}^n$ ?
d. What is $L\left(L^{+}\left(\stackrel{\to }{y}\right)\right)$ , for $\stackrel{\to }{y}$ in ${ℝ}^m$ ?
e. Find $L^{+}$ for the linear transformation $L\left(\stackrel{\to }{x}\right)=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]\stackrel{\to }{x}$