Learn Implementing Matrix Transformation in Python

Solving a Linear System

We define a system of equations:

2x + y = 5 \\ x - y = 1

This is rewritten as:


              123456789
            
import numpy as np

A = np.array([[2, 1],   # Coefficient matrix
              [1, -1]])

b = np.array([5, 1])    # Constants (RHS)

x_solution = np.linalg.solve(A, b)
print(x_solution)

This finds the values of $x$ and $y$ that satisfy both equations.

Why this matters: solving systems of equations is foundational in data science - from fitting linear models to solving optimization constraints.

Applying Linear Transformations

We define a vector:

v = np.array([[2], [3]])

Then apply two transformations:

Scaling

We stretch $x$ by 2 and compress $y$ by 0.5:

S = np.array([[2, 0],
              [0, 0.5]])

scaled_v = S @ v

This performs:

S \cdot v = \begin{bmatrix} 2 & 0 \\ 0 & 0.5 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 4 \\ 1.5 \end{bmatrix}

Rotation

We rotate the vector $90°$ counterclockwise using the rotation matrix:

theta = np.pi / 2
R = np.array([[np.cos(theta), -np.sin(theta)],
              [np.sin(theta),  np.cos(theta)]])

rotated_v = R @ v

This yields:

R \cdot v = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} -3 \\ 2 \end{bmatrix}

Visualizing the Transformations

Using matplotlib, we plot each vector from the origin, with their coordinates labeled:


              1234567891011121314151617181920212223242526272829303132333435363738394041424344454647
            
import numpy as np
import matplotlib.pyplot as plt

# Define original vector
v = np.array([[2], [3]])

# Scaling
S = np.array([[2, 0],
              [0, 0.5]])
scaled_v = S @ v

# Rotation
theta = np.pi / 2
R = np.array([[np.cos(theta), -np.sin(theta)],
              [np.sin(theta),  np.cos(theta)]])
rotated_v = R @ v

# Vizualization
origin = np.zeros(2)
fig, ax = plt.subplots(figsize=(6, 6))

# Plot original
ax.quiver(*origin, v[0], v[1], angles='xy', scale_units='xy', scale=1, color='blue', label='Original')

# Plot scaled
ax.quiver(*origin, scaled_v[0], scaled_v[1], angles='xy', scale_units='xy', scale=1, color='green', label='Scaled')

# Plot rotated
ax.quiver(*origin, rotated_v[0], rotated_v[1], angles='xy', scale_units='xy', scale=1, color='red', label='Rotated')

# Label vector tips
ax.text(v[0][0]+0.2, v[1][0]+0.2, "(2, 3)", color='blue')
ax.text(scaled_v[0][0]+0.2, scaled_v[1][0]+0.2, "(4, 1.5)", color='green')
ax.text(rotated_v[0][0]-0.6, rotated_v[1][0]+0.2, "(-3, 2)", color='red')

# Draw coordinate axes
ax.annotate("", xy=(5, 0), xytext=(0, 0), arrowprops=dict(arrowstyle="->", linewidth=1.5))
ax.annotate("", xy=(0, 5), xytext=(0, 0), arrowprops=dict(arrowstyle="->", linewidth=1.5))
ax.text(5.2, 0, "X", fontsize=12)
ax.text(0, 5.2, "Y", fontsize=12)

ax.set_xlim(-5, 5)
ax.set_ylim(-5, 5)
ax.set_aspect('equal')
ax.grid(True)
ax.legend()
plt.show()

Why this matters: data science workflows often include transformations, for example:

Principal Component Analysis - rotates data;
Feature normalization - scales axes;
Dimensionality reduction - projections.

By visualizing vectors and their transformations, we see how matrices literally move and reshape data in space.

Everything was clear?

Thanks for your feedback!

Section 4. Chapter 6

Ask AI

Ask anything or try one of the suggested questions to begin our chat

Suggested prompts:

Can you explain how the solution to the system of equations is calculated?

How does the scaling and rotation transformation affect the original vector?

Can you walk me through the visualization code and what each part does?

Awesome!

Completion rate improved to 1.96

Swipe to show menu