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Chapter 4: Rank, Triangular Factorizations, and Row Equivalence

Chapter 4: Rank, Triangular Factorizations, and Row Equivalence

pp. 74-93

Authors

, Pomona College, California, , University of Utah
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Extract

Chapter 4: In this chapter, we collect some important facts about matrices: the rank-nullity theorem; the intersection and sum of column spaces; rank inequalities for sums and products of matrices; the LU factorization and solutions of linear systems; row equivalence, the pivot column decomposition, and the reduced row echelon form. In a final capstone section, we use linear dependence, the trace, block matrices, induction, and similarity to characterize matrices that are commutators. Throughout the chapter, we emphasize block-matrix methods.

Keywords

  • Rank-nullity theorem
  • dimension of intersection
  • rank of a sum
  • rank of a product
  • LU factorization
  • row equivalence
  • Shoda’s theorem
  • trace-zero matrices
  • commutators

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