Since vectors, as n-tuples, are ordered lists of n components, it is possible to summarize and manipulate data efficiently in this framework.įor example, in economics, one can create and use, say, 8-dimensional vectors or 8-tuples to represent the Gross National Product of 8 countries. Although many people cannot easily visualize vectors in n-space, such vectors or n-tuples are useful in representing data. Most of the useful results from 2 and 3-space can be extended to these higher dimensional spaces. A vector space of dimension n is called an n-space. Modern linear algebra has been extended to consider spaces of arbitrary or infinite dimension. Vectors can be used to represent physical entities such as forces, and they can be added and multiplied with scalars, thus forming the first example of a real vector space. A vector, here, is a directed line segment, characterized by both its magnitude represented by length, and its direction. Linear algebra had its beginnings in the study of vectors in Cartesian 2-space and 3-space. Arthur Cayley introduced (2×2) matrices, one of the most fundamental linear algebraic ideas, in 1857. In 1844, Hermann Grassmann published his book Die lineale Ausdehnungslehre (see References). In 1843, William Rowan Hamilton (from whom the term vector stems) discovered the quaternions. The history of modern linear algebra dates back to the years 18.
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