Chapter R  Representations

Previous work with linear transformations may have convinced you that we can convert most questions about linear transformations into questions about systems of equations or properties of subspaces of m. In this section we begin to make these vague notions precise. We have used the word “representation” prior, but it will get a heavy workout in this chapter. In many ways, everything we have studied so far was in preparation for this chapter.

 Section VR Vector Representations
  Subsection CVS: Characterization of Vector Spaces
  Subsection CP: Coordinatization Principle
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section MR Matrix Representations
  Subsection NRFO: New Representations from Old
  Subsection PMR: Properties of Matrix Representations
  Subsection IVLT: Invertible Linear Transformations
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section CB Change of Basis
  Subsection EELT: Eigenvalues and Eigenvectors of Linear Transformations
  Subsection CBM: Change-of-Basis Matrix
  Subsection MRS: Matrix Representations and Similarity
  Subsection CELT: Computing Eigenvectors of Linear Transformations
  Subsection READ: Reading Questions
  Subsection EXC: Exercises
  Subsection SOL: Solutions
 Section OD Orthonormal Diagonalization
  Subsection TM: Triangular Matrices
  Subsection UTMR: Upper Triangular Matrix Representation
  Subsection NM: Normal Matrices
  Subsection OD: Orthonormal Diagonalization
 Section NLT Nilpotent Linear Transformations
  Subsection NLT: Nilpotent Linear Transformations
  Subsection PNLT: Properties of Nilpotent Linear Transformations
  Subsection CFNLT: Canonical Form for Nilpotent Linear Transformations
 Section IS Invariant Subspaces
  Subsection IS: Invariant Subspaces
  Subsection GEE: Generalized Eigenvectors and Eigenspaces
  Subsection RLT: Restrictions of Linear Transformations
 Section JCF Jordan Canonical Form
  Subsection GESD: Generalized Eigenspace Decomposition
  Subsection JCF: Jordan Canonical Form
  Subsection CHT: Cayley-Hamilton Theorem