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Week 8, Long Day: Linear Maps

A function $\,y=f(x)\,$ attaches to every real number $\,x\,$ a real number $\,y\,$. This is called a map!

Today’s topic is linear maps where this idea is expanded to vector inputs and outputs. The linear map $\,\mathbf y=f(\mathbf x)\,$ attaches to a vector $\,\mathbf x\,$ a new vector $\,\mathbf y\,$, which both are linear. We will investigate how typical tasks such as computing those values that provide zeroes, solving functional equations, and determining the range appear by the use of linear maps between vector spaces. In finite-dimensional vector spaces with given bases a linear transformation is represented by a mapping matrix. We will touch upon how such mapping matrix changes when we change basis, a topic that will be continued on Short Day.

Today’s Key Concepts

Linear maps. Domain and codomain. Kernel and range. Mapping matrix.

Preparation and Syllabus

Today’s content is from eNote 12 Linear Transformations up to and including Section 12.7.

Maple Ressources

  • Remember that matrices and vectors can be multiplied in sequence with a period: M.v and M.F.v and so on.

Today’s Maple demo is: KernelAndImageSpace

Activity Program

  • 10.00 – 12.00: Lecture (aud. 42, b. 303A) (link to streaming)
  • 12.30 – 17.00: Group exercises in the study areas (b. 302, bottom floor)
  • 13.00 – 16.00: Your teachers are present in the study areas

Group Exercises:

  1. The Two Linearity Requirements
  2. Investigation of a Linear Map
  3. Linear Maps in the Plane
  4. Study of Diagonal Matrices
  5. The Dimension Theorem
  6. Mapping Matrices for Reflections
  7. New Mapping Matrices by Change of Basis
  8. Polynomial Spaces