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.
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