I wrote a demo program for linear transformation of a matrix

at first

Mathematical Girl's Secret Note "What the Procession Draws" Demonstration of linear transformation performed by "Lisa" in Chapter 4 Transform I went with Python

Operating environment

Python3

Implementation example and result

Below, they are listed in the order of description in this manual.

Show dots

Displayed at point (2, 1) on the graph

import matplotlib.pyplot as plt
import numpy as np

p21 = np.array([2, 1])
plt.plot(p21[0], p21[1], marker='.')

plt.show()

Move points

Matrix points (2, 1)

\begin{pmatrix}
2 & 0 \\
0 & 2 
\end{pmatrix}

Linearly transform with to move to point (4, 2)

import matplotlib.pyplot as plt
import numpy as np

p21 = np.array([2, 1])
p21to42 = np.array([[2, 0], [0, 2]]) @ p21

fig = plt.figure()
ax = fig.add_subplot(111)
ax.annotate('', xy=p21to42,
                xytext=p21,
                arrowprops=dict(shrink=0, width=1, headwidth=8))
ax.set_xlim([0, 5])
ax.set_ylim([0, 5])

plt.plot(p21[0], p21[1], marker='.')
plt.plot(p21to42[0], p21to42[1], marker='.')

plt.show()

Move multiple points

Matrix multiple points

\begin{pmatrix}
2 & 0 \\
0 & 2 
\end{pmatrix}

Linear transformation with

import matplotlib.pyplot as plt
import numpy as np

fig = plt.figure()
ax = fig.add_subplot(111)

for x in np.linspace(-1, 1, 11):
    for y in np.linspace(-1, 1, 11):
        marker = '8'
        if x > 0 and y > 0:
            marker = '$1$'
        elif x < 0 and y > 0:
            marker = '$2$'
        elif x < 0 and y < 0:
            marker = '$3$'
        elif x > 0 and y < 0:
            marker = '$4$'

        pOrg = np.array([x, y])
        pTra = np.array([[2, 0], [0, 2]]) @ pOrg

        ax.annotate('', xy=pTra,
                        xytext=pOrg,
                        arrowprops=dict(shrink=0.1, width=1, headwidth=3))

        plt.plot(pOrg[0], pOrg[1], marker = marker)
        plt.plot(pTra[0], pTra[1], marker = marker)

ax.set_xlim([-2, 2])
ax.set_ylim([-2, 2])
plt.show()

\begin{pmatrix}
1/2 & 0 \\
0 & 1/2 
\end{pmatrix}

When linearly transformed with

\begin{pmatrix}
3 & 0 \\
0 & 2 
\end{pmatrix}

When linearly transformed with

\begin{pmatrix}
2 & 1 \\
1 & 3 
\end{pmatrix}

When linearly transformed with

Synthesis of linear transformations

Matrix of figures surrounded by (0, 0), (1, 0), (1, 1), (0, 1)

\begin{pmatrix}
2 & 1 \\
1 & 3 
\end{pmatrix}

Matrix after conversion with

\begin{pmatrix}
0 & -1 \\
1 & 0 
\end{pmatrix}

Convert with

import matplotlib.pyplot as plt
import numpy as np

fig = plt.figure()
ax1 = fig.add_subplot(131)
ax2 = fig.add_subplot(132)
ax3 = fig.add_subplot(133)

original = np.array([[0, 0], [0, 1], [1, 1], [1, 0], [0, 0]])
trans1 = np.array([[0, 0]])
trans2 = np.array([[0, 0]])
for ele in original:
    _ele1 = np.array([[2, 1], [1, 3]]) @ ele
    _ele2 = np.array([[0, -1], [1, 0]]) @ _ele1
    trans1 = np.vstack((trans1, np.array([_ele1])))
    trans2 = np.vstack((trans2, np.array([_ele2])))

ax1.plot(original[:,0], original[:,1], marker = ".", label='original')
ax2.plot(trans1[:,0], trans1[:,1], marker = ".", label='trans1')
ax3.plot(trans2[:,0], trans2[:,1], marker = ".", label='trans2')

ax1.legend(loc = 'upper center')
ax2.legend(loc = 'upper center')
ax3.legend(loc = 'upper center')
ax1.set_xlim([-4, 4])
ax1.set_ylim([-4, 4])
ax2.set_xlim([-4, 4])
ax2.set_ylim([-4, 4])
ax3.set_xlim([-4, 4])
ax3.set_ylim([-4, 4])
plt.show()

If the conversion order is different as shown in the figure below, the figure will also be different. You can see that ABx = BAx does not hold like a normal formula