7) Transformation Over Time

Transformations Over Time

1. What Does “Over Time” Mean?

When we apply a transformation repeatedly or continuously over time, the result is a dynamic system. The transformation evolves based on time (t) or steps (n).

• Example: Rotating a shape frame by frame in an animation.
• Example: A car moving along a curved path using translations and rotations.

2. Linear Transformations Over Time

A linear transformation over time follows the equation:

vt+1 = A vt

where:

• v_t is the vector at time t.
• A is the transformation matrix.
• v_t+1 is the transformed vector at t+1.

Example: Scaling Over Time

A = [ 2  0 ]
    [ 0  2 ]

Applying it repeatedly leads to exponential growth:

v1 = A v0, v2 = A v1, v3 = A v2

3. Continuous Transformations (Differential Equations)

Some transformations happen smoothly over time and are described by differential equations:

d v / dt = A v

Example: Continuous Rotation (2D Circular Motion)

A = [  0 -1 ]
    [  1  0 ]

This matrix rotates the vector infinitely in a circular motion.

4. Real-World Applications

• Physics: Particle motion, waves, and forces.
• Computer Graphics: 3D object transformations.
• Machine Learning: Neural network weight updates.
• Finance: Stock price modeling.

Next Step: Eigenvalues & Eigenvectors

Some transformations over time reveal stable patterns, where certain directions remain unchanged except for stretching/shrinking. This is the concept of eigenvalues and eigenvectors, which we will explore next.