Unit 6 of 11

Orthogonality and Projections

  • 6 lessons
  • Videos 27 to 32

Inner products bring length, angle and perpendicularity into linear algebra. This unit uses them to project vectors onto lines and subspaces and to build orthonormal bases.

The dot product gives Euclidean length, distance and angle, and a weighted inner product shows that these depend on the chosen way of measuring. The Cauchy-Schwarz and triangle inequalities follow. Projection onto a line and then onto a subspace finds the closest vector, orthogonal complements connect the four fundamental subspaces, and the Gram-Schmidt process turns independent columns into orthonormal ones, recorded as a QR factorization.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Length, Angle and Inner Products

  1. Dot Products, Length and Angle

    How the dot product of real vectors gives Euclidean length, unit vectors, the distance between vectors, a test for orthogonal vectors and the angle between nonzero vectors.

    Video 27

  2. Inner Products and Cauchy-Schwarz

    What a real inner product is, how a weighted inner product can change length and angle, and how Cauchy-Schwarz, its equality case and the triangle inequality follow from the axioms.

    Video 28

Projections and Orthogonal Complements

  1. Orthonormal Coordinates and Line Projection

    How the projection coefficient gives the projection of a vector onto a line, why the residual is perpendicular and the projection is the closest point, and how orthonormal bases and orthogonal matrices give coordinates.

    Video 29

  2. Orthogonal Complements and the Four Subspaces

    What the orthogonal complement of a subspace is, why the null space is the complement of the row space and the left null space the complement of the column space, and how complementary dimensions give orthogonal decompositions.

    Video 30

  3. Projection onto a Subspace

    How to project a vector onto a subspace with the normal equations, why the projection matrix is idempotent and symmetric, why the projection is the closest vector in the subspace, and why it does not depend on the basis.

    Video 31

Orthonormal Bases and QR

  1. Gram-Schmidt and QR Factorization

    How Gram-Schmidt turns independent vectors into orthonormal ones with the same span, how its unit vectors and coefficients form a reduced QR factorization with an upper triangular factor, and what a zero remainder reveals.

    Video 32