Q need not be a square matrix. for a non square Q matrix it’s called orthonormal matrix

If Q is a square matrix it is called orthogonal matrix

if Q is a square matrix and knowing the columns of Q are orthonormal and span the full Rn space any vector in Rn is exactly where it will be after projection hence QQt will be I

with A columns being orthonormal with respect to each other the solution to the least squares method becomes trivial as AtA becomes I

A=QR Q is a orthonormal matrix we get after doing Gram Schmidt procedure on the original basis of A (which were independent)

provided that the first vector to be chosen when doing Gram Schmidt was a1

then then q2,…..qn would all be perpendicular to a1 as those are derived from a2,…,an and while doing gram schmidt their components in the direction of a1 were removed from them

similarly it can be shown that q3…qn would be perpendicular to a2 as those are derived from a3,…,an and while doing gram schmidt their components in the direction of a2 were removed from them.

using mathematical induction we can show any column vector of the original matrix A will be perpendicular to the corresponding column vectors that come after the similar numbered column vector in Q

with the above in mind what combinations of columns of Q can give the columns of A ? for a1 only q1 is contributing as rest are perpendicular making their contribution zero