Preface to the Paradigms 4, 5, and 6: Matrices and Operators

The winter Preface is a week-long math primer in matrix methods needed for the Spins, Waves, and Central Forces paradigms. Students are assumed to have already had an introduction to linear algebra, but student who have not can still be successful. This course serves to review matrix algebra and frames the discussion of matrices in the context of vectors spaces and linear transformations. The course contains a sequence of two small group activities that emphasize rotation matrices, eigenvectors and eigenvalues, and the link between the type of transformation and the value of the determinant.

Textbook: The course content for this book is covered in any linear algebra text (except for bra-ket notation) and/or any mathematical methods text.

Sample Syllabus: Winter 2008

Course Contents

Unit: Matrices and Vectors

Matrix Manipulations (45 minutes)

Introduction to Bra-ket notation (15 minutes)

Unit: Operators and Transformations in Linear Systems

Linear Transformations (1 1/2 hr)

Tangible Metaphor for Complex Vectors (10 minutes)

Properties of Linear Vector Spaces (30 min)

Inner Products and Norms

  • Calculating Inner Products and Normalization of Vectors (Lecture, ?? minutes)

FIXME Write these lecture notes. Emphasize complex vectors.

Matrix Components (40 min)

Rotation Matrices in 2 and 3 Dimensions (10 min)

Unit: Eigenvalues and Eigenvectors

Eigenvalues & Eigenvectors (2 hr)

Special Properties of Hermitian Matrices (40 min)

Commuting Matrices (10 min)

Activities Included

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