Chapter 5.4, Problem 13E

Consider a linear transformation $L\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$ from ${ℝ}^n$ to ${ℝ}^m$ . The pseudoinverse $L^{+}$ of L is the transformation from ${ℝ}^m$ to ${ℝ}^n$ given by
$L^{+}\left(\stackrel{\to }{y}\right)=$ (the minimal least-square solution of the system $L\left(\stackrel{\to }{x}\right)=\stackrel{\to }{y}$ ).
See Exercise 8,11, and 12 for special cases.
a. Show that the transformation $L^{+}$ is linear.
b. What is $L^{+}\left(L\left(\stackrel{\to }{x}\right)\right)$ , for $\stackrel{\to }{x}$ in ${ℝ}^n$ ?
c. What is $L\left(L^{+}\left(\stackrel{\to }{y}\right)\right)$ , for $\stackrel{\to }{y}$ in ${ℝ}^m$ ?
d. Determine the image and kernel of $L^{+}$ [in terms of $im\left(A^T\right)$ and $\ker \left(A^T\right)$ ].
e. Find $L^{+}$ for the linear transformation $L\left(\stackrel{\to }{x}\right)=\left[\begin{array}{ccc}2& 0& 0\\ 0& 0& 0\end{array}\right]\stackrel{\to }{x}$ .