intuition.
all domains 22 concepts · 4 acts

Linear Algebra

The math of space, motion, and data, seen instead of memorized. Every idea here is a transformation you can picture. This is the exact spine you climb before machine learning, physics, or graphics.

Act I

The objects

6 concepts
  1. 1
    The vector

    Start with one arrow, and the decision that makes everything else possible.

  2. 2
    Adding and scaling

    Only two operations exist here. Everything later is built from them.

    needs first: vectors
  3. 3
    Linear combinations

    Scale, add, and find out how much of the plane you can cover.

    needs first: vector arithmetic
  4. 4
    Span and basis

    Everything a set of vectors can reach, and the smallest set that reaches it.

    needs first: linear combination
  5. 5
    Independence

    A test for whether a new vector is pulling its weight.

    needs first: span and basis
  6. 6
    Subspaces and dimension

    The flat worlds living inside a bigger space, and how to count their size.

    needs first: linear independence
Act II

The motions

4 concepts
  1. 7
    Transformations

    Stop moving one arrow. Move all of space at once.

    needs first: span and basis
  2. 8
    The matrix

    Why that row-times-column rule is not arbitrary at all.

    needs first: linear transformations
  3. 9
    Multiplication

    Do one motion, then another, and collapse both into a single matrix.

    needs first: the matrix
  4. 10
    Into three dimensions

    One more basis vector, and nothing else changes.

    needs first: matrix multiplication
Act III

The structure

5 concepts
  1. 11
    The determinant

    One number for how much a transformation stretches space, and whether it flips.

    needs first: linear transformations
  2. 12
    Determinant as volume

    The same number in 3D, measuring volume and catching flatness.

    needs first: determinant, transformations 3d
  3. 13
    Rank and collapse

    Two matrices that look almost identical. One loses a whole dimension.

    needs first: determinant, subspaces
  4. 14
    Running it backwards

    Solving a system is asking a motion to reverse. Sometimes it cannot.

    needs first: determinant, rank and collapse
  5. 15
    The null space

    The directions a transformation destroyed, and why solutions come in families.

    needs first: rank and collapse, inverse
Act IV

The payoff

7 concepts
  1. 16
    The dot product

    A number measuring how much two vectors agree, and a perfect test for right angles.

    needs first: linear transformations
  2. 17
    The cross product

    Two vectors in 3D, and the vector standing perpendicular to both.

    needs first: dot product, transformations 3d
  3. 18
    Change of basis

    The same arrow, described in a different language.

    needs first: inverse, span and basis
  4. 19
    Eigenvectors

    Almost every arrow gets knocked off its line. Find the ones that refuse.

    needs first: determinant, change of basis
  5. 20
    Diagonalization

    Use the eigenvectors as your axes, and the transformation becomes trivial.

    needs first: eigenvectors, change of basis
  6. 21
    Symmetric matrices

    The best-behaved matrices of all, with a guarantee you can rely on.

    needs first: diagonalization
  7. 22
    PCA: the destination

    Eigenvectors of real data, and why they are the directions that matter.

    needs first: symmetric matrices
Where this leads

Machine Learning

This was never abstract for its own sake. Eigenvectors become PCA. Linear transformations become the layers of a neural network. Every step here was already walking toward here.