The code cells below perform the following tasks:

Create matrices $A_0,$dots,$A_9$ each of size $53$ which represent the digits $0$ to $9$

Visualize the matrices as $53$ pixel images

Flatten each matrix $A_k$ into a vector $v_k$ of length $15$ and construct a $1510$ matrix A with

vectors $v_0,$dots,$v_9$ in the columns:

$A=\left[\begin{array}{ccc}?& ?& ?\\ v_0& \text{c}\text{d}\text{o}\text{t}\text{s}& v_9\end{array}\right]$

Use the matrix $A$ to answer the problems $($a$)-($e$)$ below.

$A0=np.$array$([[1.,1.,1.],$

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

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

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

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

$A1=np.$array$([[0.,1.,0.],$

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

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

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

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

$A2=np.$array$([[1.,1.,1.],$

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

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

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

$[1.,1.,1.]])A3=np.$array$([[1.,1.,1.],$

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

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

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

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

$A4=np.$array$([[1.,0.,1.],$

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

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

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

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

$A5=np.$array$([1.,1.,1.],$

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

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

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

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

$A6=np.$array$([1.,1.,1.],$

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

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

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

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

$A7=np.$array$([[1.,1.,1.],$

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

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

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

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

A$8=$ np$.$array$([[1.,1.,1.],$ Flatten each matrix into a vector of length $15$ and put the vectors into the columns of a matrix $A$ :

$A=$ np$.$column$\_$stack$([$digit$.$flatten$()$ for digit in digits$])$

A

Let $v_k$ be the $k$ th column of $A.$ Remember that indices start at $0$ in Python. The vector $v_k$

reshaped into a $53$ matrix is exactly $A_k.$ For example, the column at index $4$ in A reshaped

into a $53$ matrix is displayed below. Problem $2$c $(3$ marks$)$

The digit $8$ looks like a combination of $2$ and $5.$ Compute the orthogonal projection of $v_8$ onto

span$\{v_2,v_5\}.$ Save the result as proj$825.$

Note that you can select columns $2$ and $5$ from matrix A and save to variable

with the

command:

$A25=A[$:$,[2,5]]$

and compute the thin QR decomposition using the function scipy. linalg. qr with the

parameter mode$=$'economic':

Q$1,$R$1=$ la$.$qr$($A$25,$mode$=$'economic'$)$

Recall the projection of a vector $v$ onto $R(M)$ for some matrix $M$ is $Q_1{Q}_1^{T}v$ where

$M=Q_1{R}_1.$

#$Y$

# Test $1$: Verify type and size of proj$825.(1$ mark$)$

assert isinstance$($proj$825,$np$.$ndarray$)$

assert proj$825.$size $==15$

print$("$Test $1$: Success!"$)$plt$.$show$()$

Problem $2$d $(2$ marks$)$

Find the shortest distance from $v_9$ to span$\{v_0,v_1,v_2,v_3,v_4\}.$ Save the result as

distance$9.$ Use the function scipy. linalg. norm to compute the norm.

# YOUR CODE HERE

# Test $1$: Verify distance$9$ is a positive number close to $0\mathrm{.}7.(1$ mark$)$

assert distance$9>0$

assert np$.$round$($distance $9,1$