Problem $3$b $(2$ marks$)$

Use $Q_1$ to compute the projection of $v$ onto $R(M)$ where

$v=\left[\begin{array}{c}1\\ 2\\ 1\\ 1\\ -1\end{array}\right]$

Save the result as

# YOUR CODE HERE

# Test $2$: Verify values of proj. This cell contains hidden tests. $(1$ mark$)$

Problem $3$c $(2$ marks$)$

Find the projection matrix $P$ which projects onto $($the orthogonal complement of the

range $R(M)).$ Save the result as $P.$ Problem $3$d $(2$ marks$)$

Find the shortest distance from the vector

$w=\left[\begin{array}{c}1\\ 3\\ 1\\ 0\\ 2\end{array}\right]$

to the subspace $R(M).$ Save the result as distance$3.$ Use the function

scipy. linalg. norm to compute the norm of a matrix.

# YOUR CODE HERE

$[1.,1.,1.],$

$[0.,1.,1.],$

$[1.,0.,1.],$

$[1.,2.,1.]$

Problem $3$a $(2$ marks$)$

Compute the thin $QR$ decomposition $M=Q_1{R}_1.$ Save the result as $Q1M$ and R$1$M $.$