Date & time: Thursday, April 10, 10:30-11:20

Place: Classroom

Coverage: Penney chapter 3, class notes: Linear Transformations

Penney gives a nice narrative summary of the main ideas of chapter 3 on pp. 198-199.
Be prepared to do problems and use knowledge from the following areas.

• Linear transformations
  • Definition of a linear transformation
    Proving/checking a given transformation is linear
    See Exercises 3.1.18-19.
  • Matrix transformation
  • Matrix representation theorem
  • Not every linear transformation is a matrix transformation.
    Examples
    • Exercise 3.1.23
    • Exercise 3.1.24
  • Matrix representation of a rotation in R²
  • Properties of linear transformations
    See Exercises 3.1.26, 3.1.27.
• Matrix multiplication
  • Composition of transformations
  • Definition of matrix multiplication
    AB = [AB₁, AB₂, ..., AB_n].
  • Other characterizations of the matrix product
    • The rows of AB are the rows of A times B. [p. 154]
    • Formula (3), p. 154.
  • The composition of two matrix transformations, when defined, is the matrix transformation corresponding to the product of their matrices.
  • Properties of matrix multiplication and matrix transformations --pp. 156-157, Exercise 3.3.18
• Images
  • The image of a set S under a transformation T, denoted T(S), is the set of all vectors T(X) corresponding to X in S.
  • The image of a transformation is the image of its domain, i.e., its range.
  • The preimage, or inverse image, of a set S in the target space of a transformation T, denoted T^-1(S), is the set of all elements X in the domain such that T(X) is in S.
  • Be able to compute images of simple sets--points, line segments, rectangles, circles, ... .
  • The image of a matrix transformation is its column space. Be able to find a basis for the image of a matrix transformation.
  • Let A be an m-by-n matrix; let r be the dimension of the image of the corresponding transformation. Then r = rank of A, and the dimension of the nullspace of A is n - r. Thus n - r gives the number of dimensions lost in the image.
  • By the translation theorem, the preimage each vector in the image of a matrix transformation is a translate of the nullspace.
  • Be able to find a basis for the nullspace of a matrix transformation.
  • Rank of Products Theorem
• Inverses
  • Definitions: many-to-one, one-to-one, onto, invertible
  • Inverse Theorem, p. 167.
    Add "(e) The nullspace of  A  is {0}."
  • Definition of the inverse transformation, p. 172
  • Definition of the inverse matrix, p. 173
  • Algorithm for calculating the inverse of a matrix
  • Properties of inverse matrices: Theorems 1 and 2, pp. 176-177, Exercises 3.4.15-16.
• The LU factorization/decomposition
  • LU Decomposition Theorem
  • Algorithm for finding L and U
  • Using the factorization A = LU to solve AX = B.