Projections

Projections#

Let’s break down the concept of linear algebra projection, illustrate it with SageMath code examples, and explore visualizations to make it more intuitive.

Concept of Projection

Definitions: Projection

First recall that a subspace is a subset of a vector space that itself satisfies the properties of a vector space (e.g., closure under addition and scalar multiplication).

Projection: Given a vector v and a subspace W, the projection of v onto W is a vector p within W that is closest to v. This vector p is unique and is obtained by drawing a perpendicular line from the tip of v to the subspace W.

Why Projections?

  • Approximation: Projections help us find the “best” approximation of a vector within a smaller subspace. This is crucial in data compression, noise reduction, and solving equations.

  • Decomposition: Vectors can be decomposed into a part that lies within the subspace and a part that’s orthogonal (perpendicular) to it.

SageMath Code Examples

Let’s use SageMath to explore projections in a 2D plane.

from sage.modules.free_module_element import vector
from sage.modules.free_module import VectorSpace
from sage.plot.plot3d.shapes2 import line3d

# Define a subspace (a line)
u = vector([1, 2])
V = VectorSpace(QQ, 2)  # Create a 2-dimensional vector space over rationals
subspace = V.submodule([u])  # The line spanned by vector u

# Vector to be projected
v = vector([3, 1])

# Calculate projection
proj_v = (v.dot_product(u) / u.dot_product(u)) * u

print("Projection of v onto subspace:", proj_v)

# Plotting
show(
    point(v, color='blue', size=50) +  # Original vector
    point(proj_v, color='red', size=50) +  # Projected vector
    line3d([(0, 0, 0), (u[0], u[1], 0)], color='green') +  # Subspace line
    line3d([(v[0], v[1], 0), (proj_v[0], proj_v[1], 0)], 
           linestyle='dashed', color='purple'),  # Projection line
    aspect_ratio=1  # Make the axes have equal scales
)

Projection of v onto subspace: (1, 2)

(You can interactively rotate, zoom, and pan the above 3D plots using the mouse)

Explanation:

  1. We define a vector u that spans our subspace (the line).

  2. We define a vector v that we want to project onto the line.

  3. The formula (v.dot_product(u) / u.dot_product(u)) * u calculates the projection proj_v.