In [1]:
import matplotlib.pyplot as plt
import numpy as np
$\mathbb{R}^n$ and Vectors ¶
All vectors in the vector space has a length except $0$-vector, and the terminology of length is norm, the general formula of norm of a vector in $\mathbb{R}^k$ is given by
$$ \|u\|= \sqrt{x_1^2+x_2^2+...+x_k^2} $$As an example, let's plot vectors $(4, 7)$ and $(8, 6)$ in $\mathbb{R}^2$.
In [11]:
# Define the figure and axis
fig, ax = plt.subplots(figsize=(8, 8))
# Define the vectors as starting points and directions
vectors = np.array([[0, 0, 4, 7], [0, 0, 8, 6]])
# Unpack the vectors into separate components
X, Y, U, V = zip(*vectors)
# Plot the vectors using quiver
ax.quiver(X, Y, U, V, angles='xy', scale_units='xy', scale=1, color='red', alpha=0.6)
# Set the limits and labels for the plot
ax.set_xlim([0, 10])
ax.set_ylim([0, 10])
ax.set_xlabel('x-axis', fontsize=16)
ax.set_ylabel('y-axis', fontsize=16)
# Add grid
ax.grid()
# Annotate the end points of the vectors
ax.text(4, 7, '$(4, 7)$', fontsize=16)
ax.text(8, 6, '$(8, 6)$', fontsize=16)
# Draw lines to the origin for the first vector
ax.plot([4, 4], [0, 7], color='blue', linewidth=2)
ax.plot([0, 4], [0, 0], color='blue', linewidth=2)
# Draw lines to the origin for the second vector
ax.plot([8, 0], [0, 0], color='blue', linewidth=2)
ax.plot([8, 8], [0, 6], color='blue', linewidth=2)
# Annotate the magnitudes of the vectors using Pythagorean theorem
ax.text(1.7, 3.8, r'$\sqrt{4^2+7^2}$', fontsize=16, rotation=np.arctan(7/4)*180/np.pi)
ax.text(5, 4.1, r'$\sqrt{8^2+6^2}$', fontsize=16, rotation=np.arctan(6/8)*180/np.pi)
# Show the plot
plt.show()
There is also a NumPy function for computing norms: np.linalg.norm().
In [3]:
a = np.array([4, 7])
b = np.array([8, 6])
a_norm = np.linalg.norm(a)
b_norm = np.linalg.norm(b)
print('Norm of a and b are {0:.4f}, {1} respectively.'.format(a_norm, b_norm))
Norm of a and b are 8.0623, 10.0 respectively.
Let's try plotting vectors $(1, 1, 1)$, $(-2, 2, 5)$, $(3, -2, 1)$ in $\mathbb{R}^3$.
In [5]:
fig = plt.figure(figsize=(10,10))
ax = fig.add_subplot(111, projection='3d')
vec = np.array([[0, 0, 0, 1, 1, 1],
[0, 0, 0, -2, 2, 5],
[0, 0, 0, 3, -2, 1]])
X, Y, Z, U, V, W = zip(*vec)
ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False,
color = 'red', alpha = .6,
arrow_length_ratio = .18, pivot = 'tail',
linestyles = 'solid',linewidths = 3)
ax.set_xlim([-5, 5])
ax.set_ylim([-5, 5])
ax.set_zlim([0, 5])
ax.text(1, 1, 1, '$(1, 1, 1)$')
ax.text(-2, 2, 5, '$(-2, 2, 5)$')
ax.text(3, -2, 1, '$(3, -2, 1)$')
ax.text(0, 0, 0, '$(0, 0, 0)$')
points = np.array([[[1, 1, 1], [1, 1, 0]],
[[0, 0, 0], [1, 1, 0]],
[[3, -2, 1],[3, -2, 0]],
[[0, 0, 0],[3, -2, 0]],
[[-2, 2, 5],[-2, 2, 0]],
[[0, 0, 0],[-2, 2, 0]]])
for i in range(points.shape[0]): # loop through the 3rd axis
X, Y, Z = zip(*points[i,:,:])
ax.plot(X,Y,Z,lw = 2, color = 'b', alpha = 0.7)
ax.grid()
So the norm of vectors are quite obvious once you see their coordinates.
Vector Addition, Subtraction And Scalar Multiplication
The vector addition is element-wise operation, if we have vectors $\mathbf{u}$ and $\mathbf{v}$ addition:
$$\mathbf{u}+\mathbf{v}=\left[\begin{array}{c} u_{1} \\ u_{2} \\ \vdots \\ u_{n} \end{array}\right]+\left[\begin{array}{c} v_{1} \\ v_{2} \\ \vdots \\ v_{n} \end{array}\right]=\left[\begin{array}{c} u_{1}+v_{1} \\ u_{2}+v_{2} \\ \vdots \\ u_{n}+v_{n} \end{array}\right]$$And subtraction: $$\mathbf{u}+(-\mathbf{v})=\left[\begin{array}{c} u_{1} \\ u_{2} \\ \vdots \\ u_{n} \end{array}\right]+\left[\begin{array}{c} -v_{1} \\ -v_{2} \\ \vdots \\ -v_{n} \end{array}\right]=\left[\begin{array}{c} u_{1}-v_{1} \\ u_{2}-v_{2} \\ \vdots \\ u_{n}-v_{n} \end{array}\right]$$
And the scalar operations: $$c \mathbf{u}=c\left[\begin{array}{c} u_{1} \\ u_{2} \\ \vdots \\ u_{n} \end{array}\right]=\left[\begin{array}{c} c u_{1} \\ c u_{2} \\ \vdots \\ c u_{n} \end{array}\right]$$
Let's illustrate vector addition $\mathbf{u}+\mathbf{v}$ in both $\mathbb{R}^2$ and $\mathbb{R}^3$.
In [6]:
fig, ax = plt.subplots(figsize = (9,9))
vec = np.array([[[0, 0, 4, 7]],
[[0, 0, 8, 4]],
[[0, 0, 12, 11]],
[[4, 7, 8, 4]],
[[8, 4, 4, 7]]])
color = ['r','b','g','b','r']
for i in range(vec.shape[0]):
X,Y,U,V = zip(*vec[i,:,:])
ax.quiver(X, Y, U, V, angles='xy', scale_units='xy', color = color[i], scale=1, alpha = .6)
ax.set_xlim([0, 15])
ax.set_ylim([0, 15])
ax.set_xlabel('x-axis', fontsize =16)
ax.set_ylabel('y-axis', fontsize =16)
ax.grid()
for i in range(3):
ax.text(x = vec[i,0,2], y = vec[i,0,3], s = '(%.0d, %.0d)' %(vec[i,0,2],vec[i,0,3]), fontsize = 16)
ax.text(x= vec[0,0,2]/2, y = vec[0,0,3]/2, s= '$u$', fontsize = 16)
ax.text(x= 8, y = 9, s= '$v$', fontsize = 16)
ax.text(x= 6, y = 5.5, s= '$u+v$', fontsize = 16)
ax.set_title('Vector Addition', size = 18)
plt.show()
The illustration of $\mathbf{u} - \mathbf{v}$ is exactly equivalent to $\mathbf{u} + (-\mathbf{v})$.
In [7]:
fig, ax = plt.subplots(figsize=(9, 9))
vec = np.array([[[0, 0, 4, 7]],
[[0, 0, 8, 4]],
[[0, 0, -8, -4]],
[[0, 0, -4, 3]],
[[-8, -4, 4, 7]],
[[4, 7, -8, -4]],
[[8, 4, -4, 3]]])
color = ['r','b','b','g','r','b','g']
for i in range(vec.shape[0]):
X,Y,U,V = zip(*vec[i,:,:])
ax.quiver(X, Y, U, V, angles='xy', scale_units='xy', color = color[i], scale=1, alpha = .6)
for i in range(4):
ax.text(x = vec[i,0,2], y = vec[i,0,3], s = '(%.0d, %.0d)' %(vec[i,0,2],vec[i,0,3]), fontsize = 16)
s = ['$u$','$v$','$-v$','$u-v$']
for i in range(4):
ax.text(x = vec[i,0, 2]/2, y = vec[i,0, 3]/2, s = s[i], fontsize = 16)
ax.text(x= 5.8 , y = 5.8, s= '$u-v$', fontsize = 16)
ax.set_xlim([-9, 9])
ax.set_ylim([-9, 9])
ax.set_xlabel('x-axis', fontsize =16)
ax.set_ylabel('y-axis', fontsize =16)
ax.grid()
ax.set_title('Vector Subtraction', size = 18)
plt.show()
The real number $c$ in front of vectors are scalars, which are scaling the vectors as its name implies.
In [8]:
fig, ax = plt.subplots(figsize=(9, 9))
vec = np.array([[[0, 0, 2, 3]],
[[0, 0, 6, 9]],
[[0, 0, 2, 2]],
[[0, 0, 8, 8]]])
colors = ['r','b', 'r', 'b']
for i in range(vec.shape[0]):
X,Y,U,V = zip(*vec[i,:,:])
ax.quiver(X, Y, U, V, angles='xy', scale_units='xy',
color = color[i], scale=1, alpha = .6, zorder = -i)
s = ['$u$', '$3u$', '$v$', '$3v$']
for i in range(vec.shape[0]):
ax.text(x = vec[i,0, 2], y = vec[i, 0, 3],
s = '(%.0d, %.0d)' %(vec[i,0, 2], vec[i,0, 3]), fontsize = 16)
ax.text(x = vec[i,0, 2]/2, y = vec[i,0, 3]/2,
s = s[i], fontsize = 20)
ax.set_xlim([0, 10])
ax.set_ylim([0, 10])
ax.set_xlabel('x-axis', fontsize =16)
ax.set_ylabel('y-axis', fontsize =16)
ax.grid()
ax.set_title('Vector Multiplication', size = 18)
plt.show()
Now let's challenge ourselves for plotting in 3D.
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fig = plt.figure(figsize=(10,10))
ax = fig.add_subplot(111, projection='3d')
############################## Arrows and Texts #####################################
vec = np.array([[[0, 0, 0, 3, 4, 5]],
[[0, 0, 0, 3, 6, 2]],
[[0, 0, 0, 6, 10, 7]],
[[3, 4, 5, 3, 6, 2]],
[[3, 6, 2, 3, 4, 5]],
[[0, 0, 0, 8, 2, 2]],
[[0, 0, 0, 4, 1, 1]]])
colors = ['r','b','g','b','r','r','b']
for i in range(vec.shape[0]):
X,Y,Z, U,V,W = zip(*vec[i,:,:])
ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False,
color = colors[i], alpha = .6,arrow_length_ratio = .08, pivot = 'tail',
linestyles = 'solid',linewidths = 3)
ax.text(x = vec[i, 0, 3], y = vec[i, 0, 4], z= vec[i, 0, 5],
s = '$(%.0d, %.0d, %.0d)$'%(vec[i, 0, 3],vec[i, 0, 4],vec[i, 0, 5]))
############################## Axis #####################################
ax.grid()
ax.set_xlim([0, 10])
ax.set_ylim([0, 10])
ax.set_zlim([0, 10])
ax.set_title('Vector Addition and Scalar Multiplication', size = 16)
plt.show()
Similarly with $\mathbf{u}-\mathbf{v}$.
In [10]:
fig = plt.figure(figsize=(10,10))
ax = fig.add_subplot(111, projection='3d')
############################## Arrows and Texts #####################################
vec = np.array([[[0, 0, 0, 3, 4, 5]],
[[0, 0, 0, 3, 6, 2]],
[[0, 0, 0, -3, -6, -2]],
[[0, 0, 0, 0, -2, 3]],
[[-3, -6, -2, 3, 4, 5]],
[[3, 4, 5, -3, -6, -2]]])
for i in range(vec.shape[0]):
X,Y,Z, U,V,W = zip(*vec[i,:,:])
ax.quiver(X, Y, Z, U, V, W, length=1, normalize=False,
color = colors[i],arrow_length_ratio = .08, pivot = 'tail',
linestyles = 'solid',linewidths = 3, alpha = 1 - .15*i)
ax.text(x = vec[i, 0, 3], y = vec[i, 0, 4], z= vec[i, 0, 5],
s = '$(%.0d, %.0d, %.0d)$'%(vec[i, 0, 3],vec[i, 0, 4],vec[i, 0, 5]))
s = ['$u$','$v$', '$-v$','$u-v$']
for i in range(4):
ax.text(x = vec[i,0, 3]/2, y = vec[i,0, 4]/2, z= vec[i,0, 5]/2,
s = s[i], size = 15, zorder = i+vec.shape[0])
############################## Axis #####################################
ax.grid()
ax.set_xlim([-6, 6])
ax.set_ylim([-6, 6])
ax.set_zlim([-3, 6])
ax.set_title('Vector Subtraction', size = 18)
plt.show()
