The QR decomposition also works for matrices that are taller
than wide. You can try examples at live.sympy.org:

A=Matrix([[1,1,1],[1,2,4],[1,3,9],[1,4,16],[1,5,25]])

A

Q,R=A.QRdecomposition()

Q

R

This is handy for linear regression, because, if, like
in HW25, part 1, you only care about a low-dimensional subspace
of a high-dimensional space, then you can use QR just
on the first part of your basis corresponding to your theoretical model.
The columns of Q will still be orthonormal and have the correct span.
R will still be square and invertible, but much smaller;
R^-1 will still do the coordinate conversion you want:

Below is a QR-implemented linear regression of five (x,y) data points 
to a linear model that you can try in SymPy (live.sympy.org). It finds
that the best-fitting line for the data has y-intercept 6/5 and slope 43/50.

one=[1]*5

x=[0,5,10,15,20]

A=Matrix([one,x]).transpose()

A

Q,R=A.QRdecomposition()

Q

R

y=Matrix([[1,6,9,15,18]]).transpose()

y

c=Q.transpose()*y #this computes the dot products for the projection

c

a=R.inv()*c

a