How do they get from $Ac=0$ to $Q^tQRc=0$

Question

How do they get from $Ac=0$ to $Q^tQRc=0$ in this proof at this link in the problem 1 when they solve for b?

http://www.math.pitt.edu/~swigon/Matlab/Homework2_sols.pdf

Answer 1

Since $Q$ is an orthogonal matrix, its transpose $Q^t$ is its inverse, i.e., $\color{green}{Q^tQ=I}$, the $n\times n$ identity matrix. Multiplying on the left by $Q^t$ gives $$ Q^t(QR)c = (\color{green}{Q^tQ})\,Rc = \color{green}{I}\,Rc = Rc. $$ If we combine this with the fact that $\color{red}{Ac = 0}$, then we see $$ Rc = Q^t(QR)c = Q^t\color{red}{Ac} = Q^t\color{red}{0} = 0, $$ as desired.