1. The problem statement, all variables and given/known data Suppose that A is an matrix with linearly independent columns then A can be factored as, where Q is an matrix with orthonormal columns and R is an invertible upper triangular matrix. Show why it is Upper triangular matrix, and why the elements of the diagonal are Positive. 2. Relevant equations A=QR A= [x1 x2 ... xk] R= [r1 r2 ... rk] 3. The attempt at a solution (Sorry for my bad English, I am from Brazil) x1= (x1.u1)u1+(x1.u2)u2+....+(x1.uk)uk x2= (x2.u1)u1+(x2.u2)u2+....+(x2.uk)uk . . . xn= (xn.u1)u1+(xn.u2)u2+....+(xn.uk)uk now there is an argument, given Xi and Uj, if i<j , Uj and Xi are orthogonals. Which I do not know how to prove it. So, if this is valid: x1= (x1.u1)u1 x2= (x2.u1)u1+(x2.u2)u2 . . . xk= (xk.u1)u1+(xk.u2)u2+....+(xk.uk)uk which is triangular. Now I do not now why the product (x1.u1), (xk.uk) is always positive.