Unit 4 of 11

Linear Transformations and Coordinates

  • 6 lessons
  • Videos 18 to 23

This unit studies linear maps apart from any one matrix. It covers kernel, image and rank-nullity, change of basis, and linear maps on polynomials and matrices.

The kernel holds the inputs a map sends to zero and the image holds every output it reaches. The rank-nullity theorem says their dimensions add up to the dimension of the input space, which limits when a map can be one-to-one or onto. A fixed vector gets new coordinates in a new basis, and a fixed map gets a new matrix. For a map from a space to itself, with one basis for input and output, the old and new matrices are similar. Polynomials and matrices then become vectors too, and evaluating a polynomial or taking the trace of a matrix is a linear map with a kernel and an image.

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.

Kernel, Image and Rank-Nullity

  1. Kernel, Image and Rank-Nullity

    How the kernel and image of a linear map record lost and reached directions, why rank plus nullity equals the input dimension, and what this says about injective and surjective maps, fibers and compositions.

    Video 18

Change of Basis

  1. Change of Basis for Vector Coordinates

    How a basis matrix rebuilds a vector from its coordinates, how to solve for coordinates in an ordered basis, and how the change-of-basis matrix and its inverse convert coordinates while the vector stays fixed.

    Video 19

  2. Matrices in Different Bases

    How to find the matrix of a linear map in chosen input and output bases, why one linear operator in two bases gives similar matrices, and how a commutative diagram keeps the map separate from its matrix.

    Video 20

Vector Spaces Beyond Arrows

  1. Vector Spaces of Polynomials and Matrices

    The axioms that make a set with addition and scaling a vector space, why polynomials of degree at most two and 2-by-2 matrices qualify, and how coefficients and matrix units give them coordinates.

    Video 21

  2. Linear Maps on Polynomials and Matrices

    How evaluating a polynomial at fixed points and taking the trace of a matrix are linear maps, how basis images give their matrices, and how to find their kernel and image and check rank-nullity.

    Video 22

Vector Spaces and Coordinates Workshop

  1. Vector Spaces and Coordinates Workshop

    Practice in choosing and checking bases for the four spaces of a matrix, for coordinates in a new basis and for polynomial evaluation maps, using membership, independence and dimension counts.

    Video 23