Lecture 1: The Geometry of Linear Equations¶
Lecture 1
Jake notes: In this section we need to communicate row and column pictures [ ]; intuition for solving systems of equations in both ways [ ]; introduce foundations for invertibility in high dim spaces [ ] (even if we don’t use that language yet)
Summary¶
This lecture introduces linear algebra through the lens of solving systems of equations. We can view matrices two ways, either through the “row picture” (geometric intersection of lines/planes defined by each row-equation) and the “column picture” (linear combinations of vectors corresponding to each column).
Key Concepts¶
- Row Picture: Each equation represents a line (in 2D) or plane (in 3D). Solutions are the intersections of these structures. Below is an interactive visualization of the row picture for a 2x2 system of equations (ax + by = c and dx + ey = f). Adjust the sliders to change the coefficients and observe how the lines and their intersection point (solution) change. The matrix form of the system is also shown.
Source
import numpy as np
import matplotlib.pyplot as plt
from ipywidgets import interact, FloatSlider
import warnings
warnings.filterwarnings("ignore") # Suppress warnings for cleaner output
def plot_row_picture(a1=2.0, b1=1.0, c1=5.0, a2=1.0, b2=-1.0, c2=1.0):
# Create figure
fig, ax = plt.subplots(figsize=(6, 6))
# Define x range for plotting
x = np.linspace(-5, 5, 100)
# Plot line 1: ax + by = c
if b1 != 0:
y1 = (c1 - a1 * x) / b1
ax.plot(x, y1, 'b-', label=f'{a1:.1f}x + {b1:.1f}y = {c1:.1f}')
else:
ax.axvline(c1/a1 if a1 != 0 else 0, color='b', label=f'x = {(c1/a1):.1f}' if a1 != 0 else 'No line')
# Plot line 2: dx + ey = f
if b2 != 0:
y2 = (c2 - a2 * x) / b2
ax.plot(x, y2, 'r-', label=f'{a2:.1f}x + {b2:.1f}y = {c2:.1f}')
else:
ax.axvline(c2/a2 if a2 != 0 else 0, color='r', label=f'x = {(c2/a2):.1f}' if a2 != 0 else 'No line')
# Calculate intersection
A = np.array([[a1, b1], [a2, b2]])
b = np.array([c1, c2])
det = np.linalg.det(A)
if det != 0:
sol = np.linalg.solve(A, b)
x_sol, y_sol = sol
ax.plot(x_sol, y_sol, 'ko', markersize=8, label=f'Solution: ({x_sol:.2f}, {y_sol:.2f})')
print(f"Solution: x = {x_sol:.2f}, y = {y_sol:.2f}")
else:
print("No unique solution (parallel or coincident lines)")
# Set plot properties
ax.set_xlim(-5, 5)
ax.set_ylim(-5, 5)
ax.axhline(0, color='k', linestyle='-', alpha=0.3)
ax.axvline(0, color='k', linestyle='-', alpha=0.3)
ax.grid(True)
ax.legend()
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_title('Row Picture: Intersection of Lines')
plt.show()
# Display matrix form
print(f"Matrix form:\n$$\\begin{{bmatrix}} {a1:.1f} & {b1:.1f} \\\\ {a2:.1f} & {b2:.1f} \\end{{bmatrix}} \\begin{{bmatrix}} x \\\\ y \\end{{bmatrix}} = \\begin{{bmatrix}} {c1:.1f} \\\\ {c2:.1f} \\end{{bmatrix}}$$")
interact(plot_row_picture,
a1=FloatSlider(min=-5, max=5, step=0.1, value=2.0, description='a'),
b1=FloatSlider(min=-5, max=5, step=0.1, value=1.0, description='b'),
c1=FloatSlider(min=-10, max=10, step=0.1, value=5.0, description='c'),
a2=FloatSlider(min=-5, max=5, step=0.1, value=1.0, description='d'),
b2=FloatSlider(min=-5, max=5, step=0.1, value=-1.0, description='e'),
c2=FloatSlider(min=-10, max=10, step=0.1, value=1.0, description='f'))Column Picture: Rewrite as x * column1 + y * column2 = b, finding coefficients for vector combinations.
Basic solving via elimination: Subtract multiples of equations to isolate variables.
Sneak peek: invertibility
Tutorial: Solving a 2x2 System¶
Consider the system:
- 2x + y = 5
- x - y = 1
Step 1 (Elimination): Add the equations to eliminate y: 3x = 6 → x = 2.
Step 2: Substitute: 2(2) + y = 5 → y = 1.
In the row picture: Lines intersect at (2,1).
In the column picture: 2 * [2,1]^T + 1 * [1,-1]^T = [5,1]^T.
Python Example (Using NumPy)¶
import numpy as np
# Define matrix A and vector b
A = np.array([[2, 1], [1, -1]])
b = np.array([5, 1])
# Solve Ax = b
x = np.linalg.solve(A, b)
print("Solution: x =", x[0], ", y =", x[1])