Unit 2 of 10

Quotient Spaces and Duality

  • 5 lessons
  • Videos 8 to 12

A quotient space treats vectors that differ by an element of a fixed subspace as a single object, and the dual space collects all linear measurements of vectors.

You will construct quotient spaces, use them to capture what a map cannot detect, and prove the first isomorphism theorem, which identifies the quotient by a kernel with the image. Then you will build dual bases, pull measurements back through a map with the dual map, see why covector coordinates change by the inverse of the matrix that changes vector coordinates, and use annihilators to describe kernels and images without an inner product.

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.

Quotient Constructions

  1. Quotient Spaces

    Equivalence modulo a subspace, cosets and the quotient space, why adding and scaling cosets is well defined, and how an extended basis gives a quotient basis and the dimension of a quotient space.

    Video 8

  2. The First Isomorphism Theorem

    Fibers of a linear map, the quotient map and the induced map, the first isomorphism theorem that identifies the quotient by the kernel with the image, and polynomial evaluation as a worked case.

    Video 9

Linear Measurements and Dual Maps

  1. Dual Spaces and Dual Bases

    Linear functionals and the dual space, how the coordinate functionals of a basis form its dual basis, evaluation functionals on polynomials, and why nonzero conjugate-linear maps are not linear functionals.

    Video 10

  2. Dual Maps and Covectors

    How a linear map pulls a covector on its outputs back to its inputs, why the dual map reverses composition and has the transpose as its matrix, and why covector coordinates change by an inverse matrix.

    Video 11

  3. Annihilators and Fundamental Subspaces

    What the annihilator of a subspace is, why in finite dimensions its dimension is that of the space minus that of the subspace, and how annihilators describe a dual map's kernel and image, the row space and the left nullspace.

    Video 12