Matrices and Linearity

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In this chapter we take the first step in abstracting vectors and matrices to mathematical objects that are more than just arrays of numbers. We begin the discussion in Section ?? by introducing the multiplication of a matrix times a vector. Matrix multiplication simplifies the way in which we write systems of linear equations and is the way by which we view matrices as mappings. This latter point is discussed in Section ??.

The mappings that are produced by matrix multiplication are special and are called linear mappings. Some properties of linear maps are discussed in Section ??. One consequence of linearity is the principle of superposition that enables solutions to systems of linear equations to be built out of simpler solutions. This principle is discussed in Section ??.

In Section ?? we introduce multiplication of two matrices and discuss properties of this multiplication in Section ??. Matrix multiplication is defined in terms of composition of linear mappings which leads to an explicit formula for matrix multiplication. This dual role of multiplication of two matrices — first by formula and second as composition — enables us to solve linear equations in a conceptual way as well as in an algorithmic way. The conceptual way of solving linear equations is through the use of matrix inverses (or inverse mappings) which is described in Section ??. In this section we also present important properties of matrix inversion and a method of computation of matrix inverses. There is a simple formula for computing inverses of $2\times 2$ matrices based on determinants. The chapter ends with a discussion of determinants of $2\times 2$ matrices in Section ??.