$1)$ copy your vector norm function here

def VNorm$($v$,$p$)$:

$$ nrm $=0$

$$ if p $==$ np$.$inf:

$$ nrm $=$ np$.$max$($np$.$abs$($v$))$ # Infinity norm $($max absolute value in v$)$

$$ else:

$$ nrm $=$ np$.$sum$($np$.$abs$($v$)**$p$)**(1/$p$)$ # p$-$norm calculation

$$

$$ return$($nrm$)$

$$

$$

$$

$2)$ define a function

$$ name : QRref

$$ input : A $(2$d numpy array, size nxm$)$

$$ output : Q $(2$d numpy array, size nxn$),$ R $(2$d numpy array, size nxm$)$

$$ note$$ : A $=$ QR$$ is the QR decomposition of A

$$ note$$ : The function should work for any size rectangular array

$$ note$$ : You may assume A is full rank and n$>$m

$$ note$$ : Use reflectors to compute the decomposition

$$ note$$ : Use a conditional to check if a column $($below the diagonal$)$ is nearly zero $($check the vector norm$)$ before applying a reflector. Use a tolerance of $1$e$-15.$

$$

$$

def QRref$($A$)$:

$$ # add code here

$$ return$($Q$,$R$)$