Impara Implementing Matrix Transformation in Python

Matrices allow us to model, manipulate, and visualize linear systems and geometric transformations.

Step 1: 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.

Step 2: 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}

Step 3: Visualizing the Transformations

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


              12345678910111213141516171819202122232425262728293031323334353637383940414243444546
            
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.

Tutto è chiaro?

Grazie per i tuoi commenti!

Sezione 4. Capitolo 6

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Suggested prompts:

Can you explain how the matrix equation relates to the original system of equations?

How does the scaling and rotation affect the original vector?

Can you walk me through the visualization code step by step?

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Matrices allow us to model, manipulate, and visualize linear systems and geometric transformations.

Step 1: 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.

Step 2: 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}

Step 3: Visualizing the Transformations

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


              12345678910111213141516171819202122232425262728293031323334353637383940414243444546
            
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.

Tutto è chiaro?

Grazie per i tuoi commenti!

Sezione 4. Capitolo 6

Implementing Matrix Transformation in Python

Step 1: Solving a Linear System

Step 2: Applying Linear Transformations

Scaling

Rotation

Step 3: Visualizing the Transformations

1. What does this line define in the code?

2. What is the output of this operation?

3. What is the purpose of this rotation matrix?

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Implementing Matrix Transformation in Python

Step 1: Solving a Linear System

Step 2: Applying Linear Transformations

Scaling

Rotation

Step 3: Visualizing the Transformations

1. What does this line define in the code?

2. What is the output of this operation?

3. What is the purpose of this rotation matrix?