diff --git a/Book4_Ch03_Python_Codes/Bk4_Ch3_01.py b/Book4_Ch03_Python_Codes/Bk4_Ch3_01.py
new file mode 100644
index 0000000..d6c9906
--- /dev/null
+++ b/Book4_Ch03_Python_Codes/Bk4_Ch3_01.py
@@ -0,0 +1,52 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch3_01.py
+
+import matplotlib.pyplot as plt
+import numpy as np
+
+p_values = [0.05, 0.2, 0.5, 1, 1.5, 2, 4, 8, np.inf]
+
+x1 = np.linspace(-2.5, 2.5, num=101);
+x2 = x1;
+
+xx1, xx2 = np.meshgrid(x1,x2)
+
+fig, axes = plt.subplots(ncols=3,nrows=3,
+                         figsize=(12, 12))
+
+for p, ax in zip(p_values, axes.flat):
+    
+    if np.isinf(p):
+        zz = np.maximum(np.abs(xx1),np.abs(xx2))
+    else:
+        zz = ((np.abs((xx1))**p) + (np.abs((xx2))**p))**(1./p)
+    
+    # plot contour of Lp
+    ax.contourf(xx1, xx2, zz, 20, cmap='RdYlBu_r')
+    
+    # plot contour of Lp = 1
+    ax.contour (xx1, xx2, zz, [1], colors='k', linewidths = 2) 
+    
+    # decorations
+
+    ax.axhline(y=0, color='k', linewidth = 0.25)
+    ax.axvline(x=0, color='k', linewidth = 0.25)
+    ax.set_xlim(-2.5, 2.5)
+    ax.set_ylim(-2.5, 2.5)
+    ax.spines['top'].set_visible(False)
+    ax.spines['right'].set_visible(False)
+    ax.spines['bottom'].set_visible(False)
+    ax.spines['left'].set_visible(False)
+    ax.set_xlabel('$x_1$')
+    ax.set_ylabel('$x_2$')
+    ax.set_title('p = ' + str(p))
+    ax.set_aspect('equal', adjustable='box')
+
+plt.show()
diff --git a/Book4_Ch03_Python_Codes/Bk4_Ch3_02.py b/Book4_Ch03_Python_Codes/Bk4_Ch3_02.py
new file mode 100644
index 0000000..e67950d
--- /dev/null
+++ b/Book4_Ch03_Python_Codes/Bk4_Ch3_02.py
@@ -0,0 +1,53 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch3_02.py
+
+import numpy as np
+from matplotlib import pyplot as plt
+import seaborn as sns
+
+u = [0,0,4, 3]
+v = [0,0,-2,4]
+u_bis = [4,3,v[2],v[3]]
+w = [0,0,2,7]
+
+fig, ax = plt.subplots()
+
+plt.quiver([u[0], u_bis[0], w[0]],
+           [u[1], u_bis[1], w[1]],
+           [u[2], u_bis[2], w[2]],
+           [u[3], u_bis[3], w[3]],
+           angles='xy', scale_units='xy', 
+           scale=1, color=sns.color_palette())
+
+plt.axvline(x=0, color='grey')
+plt.axhline(y=0, color='grey')
+
+plt.text(3, 1, r'$||\vec{u}||_2$', 
+         color=sns.color_palette()[0], size=12,
+         ha='center',va='center')
+
+plt.text(3, 6, r'$||\vec{v}||_2$', 
+         color=sns.color_palette()[1], size=12,
+         ha='center',va='center')
+
+plt.text(0, 4, r'$||\vec{u}+\vec{v}||_2$', 
+         color=sns.color_palette()[2], size=12,
+         ha='center',va='center')
+
+plt.ylabel('$x_2$')
+plt.xlabel('$x_1$')
+plt.axis('scaled')
+ax.set_xticks(np.arange(-2,8 + 1))
+ax.set_yticks(np.arange(-2,8 + 1))
+ax.set_xlim(-2, 8)
+ax.set_ylim(-2, 8)
+ax.grid(linestyle='--', linewidth=0.25, color=[0.5,0.5,0.5])
+
+# reference: Essential Math for Data Science
diff --git a/Book4_Ch03_向量范数__数学要素__从加减乘除到机器学习.pdf b/Book4_Ch03_向量范数__数学要素__从加减乘除到机器学习.pdf
new file mode 100644
index 0000000..caf5b42
Binary files /dev/null and b/Book4_Ch03_向量范数__数学要素__从加减乘除到机器学习.pdf differ
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_01.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_01.py
new file mode 100644
index 0000000..068b44b
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_01.py
@@ -0,0 +1,41 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_01.py
+
+import numpy as np
+
+# 2d matrix
+A_matrix = np.matrix([[2,4],
+                      [6,8]])
+print(A_matrix.shape)
+print(type(A_matrix))
+
+# 1d array
+A_1d = np.array([2,4])
+print(A_1d.shape)
+print(type(A_1d))
+
+# 2d array
+A_2d = np.array([[2,4],
+                 [6,8]])
+print(A_2d.shape)
+print(type(A_2d))
+
+# 3d array
+A1 = [[2,4],
+      [6,8]]
+
+A2 = [[1,3],
+      [5,7]]
+
+A3 = [[1,0],
+      [0,1]]
+A_3d = np.array([A1,A2,A3])
+print(A_3d.shape)
+print(type(A_3d))
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_02.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_02.py
new file mode 100644
index 0000000..2cc94ff
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_02.py
@@ -0,0 +1,21 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_02.py
+
+import numpy as np
+
+A = np.matrix([[1,2,3],
+              [4,5,6],
+              [7,8,9]])
+
+# extract diagonal elements
+a = np.diag(A)
+
+# construct a diagonal matrix
+A_diag = np.diag(a)
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_03.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_03.py
new file mode 100644
index 0000000..16d7dbb
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_03.py
@@ -0,0 +1,24 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_03.py
+
+import numpy as np
+
+# define matrix
+A = np.matrix([[1, 2], [3, 4]])
+B = np.matrix([[2, 6], [4, 8]])
+
+# matrix addition
+A_plus_B = np.add(A,B)
+A_plus_B_2 = A + B
+
+
+# matrix subtraction
+A_minus_B = np.subtract(A,B)
+A_minus_B_2 = A - B
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_04.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_04.py
new file mode 100644
index 0000000..fd52361
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_04.py
@@ -0,0 +1,19 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_04.py
+
+import numpy as np
+
+k = 2
+X = [[1,2],
+     [3,4]]
+
+# scalar multiplication
+k_times_X = np.dot(k,X)
+k_times_X_2 = k*np.matrix(X)
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_05.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_05.py
new file mode 100644
index 0000000..7f5784e
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_05.py
@@ -0,0 +1,41 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_05.py
+
+import numpy as np
+
+# define matrix
+A = np.matrix([[1, 2], 
+               [3, 4],
+               [5, 6]])
+
+# scaler
+k = 2;
+
+# column vector c
+c = np.array([[3],
+              [2],
+              [1]])
+
+# row vector r
+r = np.array([[2,1]])
+
+# broadcasting principles
+
+# matrix A plus scalar k
+A_plus_k = A + k
+
+# matrix A plus column vector c
+A_plus_a = A + c
+
+# matrix A plus row vector r
+A_plus_r = A + r
+
+# column vector c plus row vector r
+c_plus_r = c + r
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_06.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_06.py
new file mode 100644
index 0000000..e25aa6c
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_06.py
@@ -0,0 +1,21 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_06.py
+
+import numpy as np
+
+A = np.array([[1, 2],
+              [3, 4]])
+
+B = np.array([[2, 4],
+              [1, 3]])
+
+# matrix multiplication
+A_times_B = np.matmul(A, B)
+A_times_B_2 = A@B
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_07.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_07.py
new file mode 100644
index 0000000..bf622bb
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_07.py
@@ -0,0 +1,32 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_07.py
+
+import numpy as np
+
+A = np.array([[1, 2]])
+
+B = np.array([[5, 6], 
+              [8, 9]])
+
+print(A*B)
+
+A = np.array([[1, 2]])
+
+B = np.matrix([[5, 6], 
+              [8, 9]])
+
+print(A*B)
+
+A = np.matrix([[1, 2]])
+
+B = np.matrix([[5, 6], 
+              [8, 9]])
+
+print(A*B)
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_08.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_08.py
new file mode 100644
index 0000000..99f8d14
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_08.py
@@ -0,0 +1,20 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_08.py
+
+from numpy.linalg import matrix_power as pw
+A = np.array([[1., 2.], 
+              [3., 4.]])
+
+# matrix inverse
+A_3 = pw(A,3)
+A_3_v3 = A@A@A
+
+# piecewise power
+A_3_piecewise = A**3
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_09.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_09.py
new file mode 100644
index 0000000..c32324c
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_09.py
@@ -0,0 +1,21 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_09.py
+
+import numpy as np
+
+A = np.matrix([[1,3],
+               [2,4]])
+
+print(A**2)
+
+B = np.array([[1,3],
+              [2,4]])
+
+print(B**2)
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_10.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_10.py
new file mode 100644
index 0000000..ba829f1
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_10.py
@@ -0,0 +1,19 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_10.py
+
+import numpy as np
+
+A = np.array([[1, 2],
+              [3, 4],
+              [5, 6]])
+
+# matrix transpose
+A_T = A.transpose()
+A_T_2 = A.T
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_11.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_11.py
new file mode 100644
index 0000000..5bf7fa6
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_11.py
@@ -0,0 +1,17 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_11.py
+
+from numpy.linalg import inv
+A = np.array([[1., 2.], 
+              [3., 4.]])
+
+# matrix inverse
+A_inverse = inv(A)
+A_times_A_inv = A@A_inverse
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_12.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_12.py
new file mode 100644
index 0000000..ab43f06
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_12.py
@@ -0,0 +1,21 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_12.py
+
+import numpy as np
+
+A = np.matrix([[1, 2], 
+               [3, 4]])
+
+print(A.I)
+
+B = np.array([[1, 2], 
+              [3, 4]])
+
+print(B.I)
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_13.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_13.py
new file mode 100644
index 0000000..861fb7a
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_13.py
@@ -0,0 +1,17 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_13.py
+
+import numpy as np
+A = np.array([[1, -1, 0], 
+              [3,  2, 4],
+              [-2, 0, 3]])
+
+# calculate trace of A
+tr_A = np.trace(A)
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_14.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_14.py
new file mode 100644
index 0000000..990e7f1
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_14.py
@@ -0,0 +1,21 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_14.py
+
+import numpy as np
+
+A = np.array([[1,2],
+              [3,4]])
+
+B = np.array([[5,6],
+              [7,8]])
+
+# Hadamard product
+A_times_B_piecewise = np.multiply(A,B)
+A_times_B_piecewise_V2 = A*B
diff --git a/Book4_Ch04_Python_Codes/Bk4_Ch4_15.py b/Book4_Ch04_Python_Codes/Bk4_Ch4_15.py
new file mode 100644
index 0000000..8fa92de
--- /dev/null
+++ b/Book4_Ch04_Python_Codes/Bk4_Ch4_15.py
@@ -0,0 +1,16 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch4_15.py
+
+import numpy as np
+A = np.array([[3, 1], 
+              [2, 4]])
+
+# calculate determinant of A
+det_A = np.linalg.det(A)
diff --git a/Book4_Ch04_矩阵__数学要素__从加减乘除到机器学习.pdf b/Book4_Ch04_矩阵__数学要素__从加减乘除到机器学习.pdf
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diff --git a/Book4_Ch05_矩阵乘法__数学要素__从加减乘除到机器学习.pdf b/Book4_Ch05_矩阵乘法__数学要素__从加减乘除到机器学习.pdf
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diff --git a/Book4_Ch06_Python_Codes/Bk4_Ch6_01.py b/Book4_Ch06_Python_Codes/Bk4_Ch6_01.py
new file mode 100644
index 0000000..bcfc7a2
--- /dev/null
+++ b/Book4_Ch06_Python_Codes/Bk4_Ch6_01.py
@@ -0,0 +1,33 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+
+# Bk4_Ch6_01.py
+
+import numpy as np
+
+A = np.array([[1, 2, 3, 0,  0],
+              [4, 5, 6, 0,  0],
+              [0, 0, 0, -1, 0],
+              [0, 0 ,0, 0,  1]])
+
+# NumPy array slicing
+
+A_1_1 = A[0:2,0:3]
+
+A_1_2 = A[0:2,3:]
+# A_1_2 = A[0:2,-2:]
+A_2_1 = A[2:,0:3]
+# A_2_1 = A[-2:,0:3]
+A_2_2 = A[2:,3:]
+# A_2_2 = A[-2:,-2:]
+
+# Assemble a matrix from nested lists of blocks
+
+A_ = np.block([[A_1_1, A_1_2],
+               [A_2_1, A_2_2]])
diff --git a/Book4_Ch06_Python_Codes/Bk4_Ch6_02.py b/Book4_Ch06_Python_Codes/Bk4_Ch6_02.py
new file mode 100644
index 0000000..1c53a13
--- /dev/null
+++ b/Book4_Ch06_Python_Codes/Bk4_Ch6_02.py
@@ -0,0 +1,54 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+
+# Bk4_Ch6_02.py
+
+import matplotlib.pyplot as plt 
+import numpy as np
+import seaborn as sns
+
+def plot_heatmap(x,title):
+
+    fig, ax = plt.subplots()
+    ax = sns.heatmap(x,
+                     cmap='RdYlBu_r',
+                     cbar_kws={"orientation": "horizontal"}, vmin=-1, vmax=1)
+    ax.set_aspect("equal")
+    plt.title(title)
+
+# Generate matrices A and B
+A = np.random.random_integers(0,40,size=(6,4))
+A = A/20 - 1
+
+B = np.random.random_integers(0,40,size=(4,3))
+B = B/20 - 1
+
+# visualize matrix A and B
+plot_heatmap(A,'A')
+
+plot_heatmap(B,'B')
+
+# visualize A@B
+C = A@B
+plot_heatmap(C,'C = AB')
+
+C_rep = np.zeros_like(C)
+
+# reproduce C
+
+for i in np.arange(4):
+    C_i = A[:,[i]]@B[[i],:];
+    title = 'C' + str(i + 1)
+    plot_heatmap(C_i,title)
+    
+    C_rep = C_rep + C_i
+
+# Visualize reproduced C
+plot_heatmap(C_rep,'C reproduced')
+
diff --git a/Book4_Ch06_Python_Codes/Bk4_Ch6_03.py b/Book4_Ch06_Python_Codes/Bk4_Ch6_03.py
new file mode 100644
index 0000000..1fd7910
--- /dev/null
+++ b/Book4_Ch06_Python_Codes/Bk4_Ch6_03.py
@@ -0,0 +1,50 @@
+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+
+# Bk4_Ch6_03.py
+
+import numpy as np
+from matplotlib import pyplot as plt
+import seaborn as sns
+
+A = np.array([[-1,  1],
+              [0.7, -0.4]])
+
+B = np.array([[0.5, -0.6],
+             [-0.8, 0.3]])
+
+A_kron_B = np.kron(A, B)
+
+fig, axs = plt.subplots(1, 5, figsize=(12, 5))
+
+plt.sca(axs[0])
+ax = sns.heatmap(A,cmap='RdYlBu_r',vmax = 1,vmin = -1,
+                 cbar_kws={"orientation": "horizontal"})
+ax.set_aspect("equal")
+plt.title('A')
+
+plt.sca(axs[1])
+plt.title('$\otimes$')
+plt.axis('off')
+
+plt.sca(axs[2])
+ax = sns.heatmap(B,cmap='RdYlBu_r',vmax = 1,vmin = -1,
+                 cbar_kws={"orientation": "horizontal"})
+ax.set_aspect("equal")
+plt.title('B')
+
+plt.sca(axs[3])
+plt.title('=')
+plt.axis('off')
+
+plt.sca(axs[4])
+ax = sns.heatmap(A_kron_B,cmap='RdYlBu_r',vmax = 1,vmin = -1,
+                 cbar_kws={"orientation": "horizontal"})
+ax.set_aspect("equal")
+plt.title('C')
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