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Linear Algebra

Build real geometric intuition for vectors, matrices and transformations — then the machinery that makes them useful.

10 chapters · 32 skills

Map what you already know

  1. Chapter 1

    Vectors and geometric intuition

    • Vectors and geometric intuition
    • Vector additionNeeds Vectors and geometric intuition
    • Scalar multiplicationNeeds Vectors and geometric intuition
    • Magnitude and unit vectorsNeeds Scalar multiplication

  2. Chapter 2

    Systems of equations

    • Linear equations as linesNeeds Vectors and geometric intuition
    • Solving systemsNeeds Linear equations as lines
    • One, none, or infinitely manyNeeds Solving systems

  3. Chapter 3

    Matrices

    • What a matrix isNeeds Solving systems
    • Matrix–vector multiplicationNeeds What a matrix is, Vector addition
    • Matrix multiplicationNeeds Matrix–vector multiplication
    • Identity and inverseNeeds Matrix multiplication

  4. Chapter 4

    Linear transformations

    • Linear transformationsNeeds Matrix–vector multiplication
    • Matrices as transformationsNeeds Linear transformations
    • Composing transformationsNeeds Matrices as transformations, Matrix multiplication

  5. Chapter 5

    Span and independence

    • Linear combinationsNeeds Scalar multiplication, Vector addition
    • SpanNeeds Linear combinations
    • Linear independenceNeeds Span

  6. Chapter 6

    Basis and dimension

    • BasisNeeds Linear independence
    • DimensionNeeds Basis
    • Rank and null spaceNeeds Dimension, Matrices as transformations

  7. Chapter 7

    Determinants

    • The determinant as areaNeeds Matrices as transformations
    • Computing determinantsNeeds The determinant as area
    • Determinants and invertibilityNeeds Computing determinants, Identity and inverse

  8. Chapter 8

    Eigenvalues and eigenvectors

    • Eigenvectors, intuitivelyNeeds Matrices as transformations
    • The characteristic equationNeeds Eigenvectors, intuitively, Computing determinants
    • DiagonalizationNeeds The characteristic equation, Basis

  9. Chapter 9

    Orthogonality

    • The dot productNeeds Magnitude and unit vectors
    • ProjectionNeeds The dot product
    • Orthogonal basesNeeds Projection, Basis

  10. Chapter 10

    Applications

    • Least squares fittingNeeds Projection, Rank and null space
    • Principal components, intuitivelyNeeds Diagonalization, Orthogonal bases
    • Transformations in graphicsNeeds Composing transformations