Abstract
This chapter discusses Euclidean space and linear mappings. Introductory calculus deals mainly with real-valued functions of a single variable. Multivariable calculus deals in general, and in a somewhat similar way, with mappings from one Euclidean space to another. However, a number of new and interesting phenomena appear resulting from the rich geometric structure of n-dimensional Euclidean space. A real-life problem can lead to a high-dimensional mathematical model. The modern techniques of automatic computation render feasible the numerical solution of many high-dimensional problems, whose manual solution can require an inordinate amount of computation. The dimension of a finite-dimensional vector space is the largest number of linearly independent vectors that it contains. An infinite-dimensional vector space is one that contains n linearly independent vectors for every positive integer n. Any system of homogeneous linear equations, with more unknowns than equations, has a nontrivial solution.
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