Lab 3.2

Interpretable Linear Transformations

Lab goal

Read a transformation matrix as an action on vectors and distinguish what rotation, projection, and reflection preserve or remove.

Lecture-note concepts

Code extensionThe full notebook applies many angles to a point cloud and extends projection to three dimensions.

Key ideas

Matrix as a mapA[x₁, x₂]ᵀ = x₁a₁ + x₂a₂: The columns of a matrix show where the standard basis vectors move; every other vector follows by linearity.
ProjectionP = uuᵀ for a unit vector u: Projection keeps the component along a chosen subspace and removes the perpendicular component.
ReflectionH = 2P − I: Reflection across span(u) keeps the projected component and reverses the perpendicular component.

Lab 3.2 · Guided example

Random point clouds are replaced by one vector, so each matrix entry can be connected directly to a visible geometric change.

Select a line to see its action, array shape, and concrete operation.

Simplified fromCh3-2 Interpretable Linear Transformations.ipynb ↗The full source visualizes larger point sets; this example isolates one mechanism at a time.

L01 / 14Choose one vector and angle

line action

import numpy as np

All traced lines

Predict

After a 90° counterclockwise turn, which quadrant contains v?

Before

The same vector v = [2,1] will make rotation, projection, and reflection directly comparable.

v = 2e₁ + 1e₂

After

Reveal to compare.

SOURCE

v.shapenot stored ?A NumPy 1-D vector.

thetanot stored ?A 90° angle in radians.

Continue in Python

Run the original code and change the inputs.