Main Theorem of QR Factorization

Proposition 2.5.1. If with linearly independent column vectors (=1:n) can be factored into A=QR, where and then
 
In particular, if

then
 

 
and
 

Proof. If A=QR, then

or

Thus, and Since rank(A)=n, then and relation (3) holds. Relation (3) for k=n yields relation (4), and this yields (5). From

it results assertion (6).