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 {ℂ}^{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
  Annotated Acronyms R: Representations