# Vectors, Pure and Applied

### Discovery module overview

This module introduces students to three outstandingly influential developments from 19th century mathematics: Complex numbers are the natural setting for much pure and applied mathematics, and vectors provide the natural language to describe mechanics, gravitation and electromagnetism, while the rigorous notion of limit is fundamental to calculus. Along the way, students will go beyond the straightforward calculation and problem solving skills emphasized in A-level Mathematics, and learn to formulate rigorous mathematical proofs.

On completion of this module, students should be able to: For nonlinear systems , which cannot be modeled with linear algebra, linear algebra is often used as a first-order approximation. The procedure for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Its use is illustrated in eighteen problems, with two to five equations. In fact, in this new geometry, now called Cartesian geometry , lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations.

The first systematic methods for solving linear systems used determinants , first considered by Leibniz in In , Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's Rule. Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy. In Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra.

In , James Joseph Sylvester introduced the term matrix , which is Latin for womb.

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Linear algebra grew with ideas noted in the complex plane. The segments are equipollent. Arthur Cayley introduced matrix multiplication and the inverse matrix in , making possible the general linear group. The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object.

## Linear algebra - Wikipedia

He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants". The telegraph required an explanatory system, and the publication of A Treatise on Electricity and Magnetism instituted a field theory of forces and required differential geometry for expression.

Linear algebra is flat differential geometry and serves in tangent spaces to manifolds. Electromagnetic symmetries of spacetime are expressed by the Lorentz transformations , and much of the history of linear algebra is the history of Lorentz transformations. The first modern and more precise definition of a vector space was introduced by Peano in ; [5] by , a theory of linear transformations of finite-dimensional vector spaces had emerged.

Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra.

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Cambridge Core - Algebra - Vectors, Pure and Applied - by T. W. Körner. Editorial Reviews. Review. "This book will be very useful for mathematics students. Also Vectors, Pure and Applied - Kindle edition by T. W. Körner. Download.

The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. Until 19th century, linear algebra was introduced through systems of linear equations and matrices. In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic, more general not limited to the finite-dimensional case , and conceptually simpler, although more abstract.

A vector space over a field F often the field of the real numbers is a set V equipped with two binary operations satisfying the following axioms. Elements of V are called vectors , and elements of F are called scalars. The second operation, scalar multiplication , takes any scalar a and any vector v and outputs a new vector av. The axioms that addition and scalar multiplication must satisfy are the following in the list below, u , v and w are arbitrary elements of V , and a and b are arbitrary scalars in the field F.

The first four axioms mean that V is an abelian group under addition. Elements of a vector space may have various nature; for example, they can be sequences , functions , polynomials or matrices. Linear algebra is concerned with properties common to all vector spaces.

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Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field F , a linear map also called, in some contexts, linear transformation, linear mapping or linear operator is a map. This implies that for any vectors u , v in V and scalars a , b in F , one has. When a bijective linear map exists between two vector spaces that is, every vector from the second space is associated with exactly one in the first , the two spaces are isomorphic.

Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in the sense that they cannot be distinguished by using vector space properties. An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range or image and the set of elements that are mapped to the zero vector, called the kernel of the map.

## Linear algebra

All these questions can be solved by using Gaussian elimination or some variant of this algorithm. The study of subsets of vector spaces that are themselves vector spaces for the induced operations is fundamental, similarly as for many mathematical structures. These subsets are called linear subspaces. These conditions suffices for implying that W is a vector space. For example, the image of a linear map, and the inverse image of 0 by a linear map called kernel or null space are linear subspaces.

Another important way of forming a subspace is to consider linear combinations of a set S of vectors: The span of S is also the intersection of all linear subspaces containing S.

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In other words, it is the smallest for the inclusion relation linear subspace containing S. A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set S of vector is linearly independent if the only way to express the zero vector as a linear combination of elements of S is to take zero for every coefficient a i. A set of vectors that spans a vector space is called a spanning set or generating set.

### Assessment and teaching

If a spanning set S is linearly dependent that is not linearly independent , then some element w of S is in the span of the other elements of S , and the span would remain the same if one remove w from S. One may continue to remove elements of S until getting a linearly independent spanning set.

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Such a linearly independent set that spans a vector space V is called a basis of V. The importance of bases lies in the fact that there are together minimal generating sets and maximal independent sets. Any two bases of a vector space V have the same cardinality , which is called the dimension of V ; this is the dimension theorem for vector spaces. Moreover, two vector spaces over the same field F are isomorphic if and only if they have the same dimension. If any basis of V and therefore every basis has a finite number of elements, V is a finite-dimensional vector space.

Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps. Their theory is thus an essential part of linear algebra. Let V be a finite-dimensional vector space over a field F , and v 1 , v 2 , By definition of a basis, the map. Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector.

It follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing exactly the same concepts. Two matrices that encode the same linear transformation in different bases are called similar. Equivalently, two matrices are similar if one can transform one in the other by elementary row and column operations. For a matrix representing a linear map from W to V , the row operations correspond to change of bases in V and the column operations correspond to change of bases in W.

Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns.

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In terms of vector space, this means that, for any linear map from W to V , there are bases such that a part of the basis of W is mapped bijectively on a part of the basis of V , and that the remaining basis elements of W , if any, are mapped to zero this is a way of expressing the fundamental theorem of linear algebra. Gaussian elimination is the basic algorithm for finding these elementary operations, and proving this theorem.

Systems of linear equations form a fundamental part of linear algebra.

Historically, linear algebra and matrix theory has been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems. Dispatched from the UK in 1 business day When will my order arrive? Home Contact Us Help Free delivery worldwide. Vectors, Pure and Applied: Description Many books in linear algebra focus purely on getting students through exams, but this text explains both the how and the why of linear algebra and enables students to begin thinking like mathematicians.

The author demonstrates how different topics geometry, abstract algebra, numerical analysis, physics make use of vectors in different ways and how these ways are connected, preparing students for further work in these areas. The book is packed with hundreds of exercises ranging from the routine to the challenging. Sketch solutions of the easier exercises are available online.

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