diff --git a/Book4_Ch01_向量运算__数学要素__从加减乘除到机器学习.pdf b/Book4_Ch01_向量运算__数学要素__从加减乘除到机器学习.pdf
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diff --git a/Book4_Ch02_Python_Codes/Bk4_Ch2_01.py b/Book4_Ch02_Python_Codes/Bk4_Ch2_01.py
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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch2_01.py
+
+import numpy as np
+
+a = np.array([[4, 3]])
+b = np.array([[5, -2]])
+
+a_dot_b = np.inner(a, b)
+
+a_2 = np.array([[4], [3]])
+b_2 = np.array([[5], [-2]])
+a_dot_b_2 = a_2.T@b_2
diff --git a/Book4_Ch02_Python_Codes/Bk4_Ch2_02.py b/Book4_Ch02_Python_Codes/Bk4_Ch2_02.py
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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch2_02.py
+
+import numpy as np
+a = np.array([[2,3],
+              [3,4]])
+
+b = np.array([[3,4],
+              [5,6]])
+
+a_@_b = np.dot(a,b)
+# a@b
diff --git a/Book4_Ch02_Python_Codes/Bk4_Ch2_03.py b/Book4_Ch02_Python_Codes/Bk4_Ch2_03.py
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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch2_03.py
+
+import numpy as np
+a = np.array([[1,2],
+              [3,4]])
+
+b = np.array([[3,4],
+              [5,6]])
+
+a_dot_b = np.vdot(a,b)
+# [1,2,3,4]*[3,4,5,6].T
diff --git a/Book4_Ch02_Python_Codes/Bk4_Ch2_04.py b/Book4_Ch02_Python_Codes/Bk4_Ch2_04.py
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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch2_04.py
+
+import numpy as np
+
+a, b = np.array([[4], [3]]), np.array([[5], [-2]])
+
+# calculate cosine theta
+cos_theta = (a.T @ b) / (np.linalg.norm(a,2) * np.linalg.norm(b,2))
+
+# calculate theta in radian
+cos_radian = np.arccos(cos_theta)
+
+# convert radian to degree
+cos_degree = cos_radian * ((180)/np.pi)
diff --git a/Book4_Ch02_Python_Codes/Bk4_Ch2_05.py b/Book4_Ch02_Python_Codes/Bk4_Ch2_05.py
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+++ b/Book4_Ch02_Python_Codes/Bk4_Ch2_05.py
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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch2_05.py
+
+from scipy.spatial import distance
+from sklearn import datasets
+import numpy as np
+
+# import the iris data
+iris = datasets.load_iris()
+
+# Only use the first two features: sepal length, sepal width
+X = iris.data[:, :]
+
+# Extract 4 data points
+x1_data = X[0,:]
+x2_data = X[1,:]
+x51_data = X[50,:]
+x101_data = X[100,:]
+
+# calculate cosine distance
+x1_x2_cos_dist = distance.cosine(x1_data,x2_data)
+x1_norm = np.linalg.norm(x1_data) 
+x2_norm = np.linalg.norm(x2_data)
+x1_dot_x2 = x1_data.T@x2_data
+x1_x2_cos = x1_dot_x2/x1_norm/x2_norm
+
+
+x1_x51_cos_dist = distance.cosine(x1_data,x51_data)
+
+x1_x101_cos_dist = distance.cosine(x1_data,x101_data)
diff --git a/Book4_Ch02_Python_Codes/Bk4_Ch2_06.py b/Book4_Ch02_Python_Codes/Bk4_Ch2_06.py
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+++ b/Book4_Ch02_Python_Codes/Bk4_Ch2_06.py
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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch2_06.py
+
+import numpy as np
+a = np.array([-2, 1, 1])
+b = np.array([1, -2, -1])
+# a = [-2, 1, 1]
+# b = [1, -2, -1]
+
+# calculate cross product of row vectors
+a_cross_b = np.cross(a, b)
+
+a_col = np.array([[-2], [1], [1]])
+b_col = np.array([[1], [-2], [-1]])
+
+# calculate cross product of column vectors
+a_cross_b_col = np.cross(a_col,b_col,axis=0)
diff --git a/Book4_Ch02_Python_Codes/Bk4_Ch2_07.py b/Book4_Ch02_Python_Codes/Bk4_Ch2_07.py
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+++ b/Book4_Ch02_Python_Codes/Bk4_Ch2_07.py
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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch2_07.py
+
+import numpy as np
+a = np.array([-2, 1, 1])
+b = np.array([1, -2, -1])
+# a = [-2, 1, 1]
+# b = [1, -2, -1]
+
+
+# calculate element-wise product of row vectors
+a_times_b = np.multiply(a, b)
+a_times_b_2 = a*b
+
+a_col = np.array([[-2], [1], [1]])
+b_col = np.array([[1], [-2], [-1]])
+
+# calculate element-wise product of column vectors
+a_times_b_col = np.multiply(a_col,b_col)
+a_times_b_col_2 = a_col*b_col
diff --git a/Book4_Ch02_Python_Codes/Bk4_Ch2_08.py b/Book4_Ch02_Python_Codes/Bk4_Ch2_08.py
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+++ b/Book4_Ch02_Python_Codes/Bk4_Ch2_08.py
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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+# Bk4_Ch2_08.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)
+
+a = np.array([[0.5],[-0.7],[1],[0.25],[-0.6],[-1]])
+b = np.array([[-0.8],[0.5],[-0.6],[0.9]])
+
+a_outer_b = np.outer(a, b)
+a_outer_a = np.outer(a, a)
+b_outer_b = np.outer(b, b)
+
+# Visualizations
+plot_heatmap(a,'a')
+
+plot_heatmap(b,'b')
+
+plot_heatmap(a_outer_b,'a outer b')
+
+plot_heatmap(a_outer_a,'a outer a')
+
+plot_heatmap(b_outer_b,'b outer b')