Linear Maps and Changes of Coordinates

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The first section in this chapter, Section ??, defines linear mappings between abstract vector spaces, shows how such mappings are determined by their values on a basis, and derives basic properties of invertible linear mappings.

The notions of row rank and column rank of a matrix are discussed in Section ?? along with the theorem that states that these numbers are equal to the rank of that matrix.

Section ?? discusses the underlying meaning of similarity — the different ways to view the same linear mapping on $\R ^n$ in different coordinates systems or bases. This discussion makes sense only after the definitions of coordinates corresponding to bases and of changes in coordinates are given and justified. In Section ??, we discuss the matrix associated to a linearity transformation between two finite dimensional vector spaces in a given set of coordinates and show that changes in coordinates correspond to similarity of the corresponding matrices.