Bloom Scroll

The mathematics of change and signals

How do transformations, rotation and frequency explain systems that change?

13 videos, about 4 h 48 min. Watch in order, or jump to a timestamp.

  1. 1

    Introductory, Visual intuition, 14 min

    Dot products and duality | Chapter 9, Essence of linear algebra

    • Relate a dot product to projection.
    • Understand dot products through linear transformations.
    • Connect vectors with their dual transformations.

    Jump to 0:34 why transformations matter, 1:30 dot products as projection

  2. 2

    Intermediate, Visual intuition, 17 min

    Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra

    • Find vectors that remain on their own span.
    • Interpret eigenvalues as stretch or squish factors.
    • Use an eigenbasis to simplify a transformation.

    Jump to 0:51 prerequisites for eigenvectors, 1:39 most vectors leave their span

  3. 3

    Advanced, Lecture, 44 min

    A Swift Introduction to Geometric Algebra

    • Introduce bivectors and higher-dimensional primitives.
    • Generalize the geometric product.
    • Interpret multiplication as rotation.

    Jump to 4:32 from scalars and vectors to bivectors, 7:49 k-vectors and the geometric product

  4. 4

    Intermediate, Visual intuition, 32 min

    Visualizing the 4d numbers Quaternions

    • Treat quaternions as a four-dimensional extension of complex numbers.
    • Visualize their behaviour in three dimensions.
    • Connect quaternion history and modern uses.

    Jump to 0:21 quaternions as four-dimensional numbers, 1:42 hamilton’s push for quaternions

  5. 5

    Introductory, Visual intuition, 27 min

    Differential equations, a tourist's guide | DE1

    • Describe a system through a rule for change.
    • Represent a damped pendulum as a trajectory.
    • Use phase space to reason about multi-variable systems.

    Jump to 13:04 a damped pendulum spirals inward, 18:03 phase space for several bodies

  6. 6

    Intermediate, Derivation, 4 min

    e^(iπ) in 3.14 minutes, using dynamics | DE5

    • Define exponential behaviour through a relation between position and change.
    • Interpret multiplication by i as a quarter turn.
    • Connect circular motion to Euler’s identity.

    Jump to 0:20 position and velocity, 1:50 what happens with the constant i

  7. 7

    Intermediate, Derivation, 27 min

    How (and why) to raise e to the power of a matrix | DE6

    • Recognize that matrix exponentiation is not repeated multiplication of e.
    • Define exponentiation through a power series.
    • Connect matrix exponentials to systems of change.

    Jump to 0:22 why matrix exponents seem nonsensical, 1:24 what the definition is not

  8. 8

    Introductory, Visual intuition, 20 min

    But what is the Fourier Transform? A visual introduction.

    • Decompose a sound into frequency components.
    • Wind a signal around a circle at different frequencies.
    • Use the centre of mass to detect matching frequencies.

    Jump to 0:27 decomposing frequencies from sound, 3:47 winding a signal around a circle

  9. 9

    Intermediate, Derivation, 25 min

    But what is a Fourier series? From heat flow to drawing with circles | DE4

    • Build complicated behaviour from rotating vectors.
    • Treat sine-wave decomposition as a special case.
    • Use linearity to solve the heat equation with simple solutions.

    Jump to 1:31 drawing with rotating vectors, 2:11 context in the heat-equation series

  10. 10

    Introductory, Visual intuition, 23 min

    But what is a convolution?

    • Introduce convolution as a new way to combine lists.
    • Find probabilities for sums by pairing outcomes.
    • Relate convolution to image processing.

    Jump to 0:20 a new way to combine lists, 3:04 flipping and sliding dice outcomes

  11. 11

    Intermediate, Visual intuition, 18 min

    The intuition behind Fourier and Laplace transforms I was never taught in school

    • Interpret transform integrals as real and imaginary areas.
    • Relate frequency ideas to exponential behaviour.
    • Compare the questions answered by Fourier and Laplace transforms.

    Jump to 1:00 integrals as real and imaginary areas, 13:52 connecting to the laplace transform

  12. 12

    Intermediate, Visual intuition, 20 min

    What does the Laplace Transform really tell us? A visual explanation (plus applications)

    • See how a pure cosine appears after transforming.
    • Interpret transformed derivatives algebraically.
    • Use transforms to analyze differential equations.

    Jump to 1:40 transforming a pure cosine, 14:36 laplace transform of a second derivative

  13. 13

    Introductory, How it works inside, 16 min

    Everything You Need to Know About Control Theory

    • Capture system dynamics in a mathematical model.
    • Use feedback to correct deviations.
    • Change system dynamics and stability through control.

    Jump to 3:47 modelling system dynamics, 6:21 feedback control