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Book 1A

Linear Algebra and the Derivative

The two languages everything later is written in

Vectors, linear maps, eigenvalues and the spectral theorem, then the derivative as the best linear approximation, in one and several variables. Computational fluency first; the foundations come in Course 2.

  • Main companionLinear Algebra Done Right · Sheldon Axler
  • Main companionCalculus · Michael Spivak
In preparation

Contents

  1. 1
    Vectors and Vector Spaces

    Colour, sound and the solutions of an equation are all vectors; what a basis is and why coordinates depend on it.

    In preparation

  2. 2
    Linear Maps and Matrices

    Matrices as the coordinate form of linear maps: composition, kernel, image and the rank–nullity theorem.

    In preparation

  3. 3
    Determinants and Volume

    The determinant as signed volume scaling, from the shoelace formula to the Jacobian.

    In preparation

  4. 4
    Eigenvalues and Repeated Maps

    What a matrix does when applied again and again: eigenvectors, diagonalisation, PageRank and population growth.

    In preparation

  5. 5
    Inner Products and Least Squares

    Length, angle and orthogonal projection, and why fitting a line to data is a projection.

    In preparation

  6. 6
    Symmetric Matrices and the Spectral Theorem

    Quadratic forms, positive-definiteness and principal axes: the first appearance of curvature’s algebra.

    In preparation

  7. 7
    The Derivative as a Linear Approximation

    Derivatives, Taylor polynomials and linearisation, from the pendulum to Newton’s method.

    In preparation

  8. 8
    Gradient, Jacobian and Hessian

    Calculus in several variables as a working tool: chain rule, gradients and the second-derivative test.

    In preparation

  9. 9
    Multiple Integrals and Change of Variables

    Integrating over regions, polar and spherical coordinates, and the Gaussian integral.

    In preparation

  10. 10
    Divergence, Curl and the Integral Theorems

    Flux, circulation and the theorems of Green, Gauss and Stokes, and how conservation laws give the heat equation.

    In preparation

  11. 11
    Linear Differential Equations

    Oscillators, circuits and cooling: solving x′ = Ax with eigenvalues.

    In preparation

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