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Linear Algebra is the study of vectors and vector spaces, their properties, and structures, and processes that are extensions of these two basic concepts. It is a part of both abstract algebra and analysis, through its extension to linear spaces. Furthermore, it has a concrete representation in analytic geometry. Finally, linear algebra has extensive applications in the pure sciences and the social sciences. Linear algebra had its foundation in the study of vectors in real 2-space and 3-space. A vector introduced the concept of directed line segment, which had direction in addition to the usual properties of a line segment like length, position, and slope. Vectors could be used then to represent certain physical entities such as forces, and the result of combining two or more forces. Linear algebra today has been extended to consider n-space, since most of the useful results from 2 and 3-space can be extended to n-dimensional space. Although one cannot visualize vectors in n-space, such vectors or n-tuples are useful in representing data. Since vectors, as n-tuples, are ordered lists of n components, one can summarize and manipulate data efficiently in this framework. For example, in economics, one can create and use, say, 8-dimensional vectors or 8-tuples to represent the [Gross National Product]? of 8 countries. One can decide to display the GNP of 8 countries for a particular year, where the countries' order is specified, for example, (United States, United Kingdom, France, Germany, Spain, India, Japan, Australia), by using a vector (v1, v2, v3, v4, v5, v6, v7, v8) where each country's GNP is in its respective position. Finally, linear algebra has its own place in mathematics. A vector space, as a purely abstract concept about which we prove properties and theorems, is part of abstract algebra, and well integrated into this field. Some striking examples of this are the group of invertible linear maps or matrices, and the ring of linear maps of a vector space. On the other side of mathematics, linear algebra plays an important part in Analysis, notably, in the study of tensor products and alternative maps. But, to begin at the beginning, one has to define some "elementary" objects and properties on which linear algebra is built and look at some examples. Included here are: *Vector space */Subspace? *[/Linear Combination]? *[/Generating a Vector Space]? *[/Linearly Independent Vectors]? *[/Basis for a Vector Space]? *[/Dimension of a Vector Space]? *Normed vector space *Inner product space *Banach space *Hilbert space A linear (vector) space is defined over a field, such as the field of real numbers or the field of complex numbers. Linear operators take elements from a linear space to another (or to itself). The set of all such transformations is itself a linear space. Associated with a linear transform is a table of numbers called a matrix. Surprisingly enough, studying the properties of matrices can be very enlightening. /Talk? |
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#REDIRECT Linear algebra |
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