Cosets and the quotient space any vector space is an abelian group under the operation of vector addition. The space obtained is called a quotient space and is denoted v n read v mod n or v by n. These notes of linear algebra course emphasize the mathematical rigour over the applications, contrary to many books on linear algebra for engineers. The quotient vector space suppose v is a vector space over k and u. Linear algebra ii lecture notes pdf 61p this book explains the following topics related to linear algebra. However, even if you have not studied abstract algebra, the idea of a coset in a vector. The authors emphasis on vector spaces over general fields, with corresponding current applications, sets the book apart. In particular, at the end of these notes we use quotient spaces to give a simpler proof than the one given in the book of the fact that operators on nite dimensional complex vector spaces are \uppertriangularizable. Linear spaces this chapter is a brief survey of basic linear algebra. Morozov itep, moscow, russia abstract concise introduction to a relatively new subject of non linear algebra. Major topics of linear algebra are presented in detail, with proofs of important theorems provided. Just knowing the open sets in a topological space can make the. The focus here is on gaining an intuition for the concepts of inverse matrices, column space, rank and null space, but the computation.
Below well provide a construction which starts with a vector space v over a eld f and a subspace sof v, and which furnishes with an entirely new vector space from vswhich is particularly prominent in applications. Quotient spaces in all the development above we have created examples of vector spaces primarily as subspaces of other vector spaces. Namely, any basis of the subspace u may be extended to a basis of the whole space v. Vector space, subspace, linear span, quotient space, linear dependence and independence and their basic properties, basis and dimension. So linear algebra studies these linear spaces and the maps between them that are compatible with the linear structure.
When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Linear algebra ii lecture notes pdf 61p download book. X is denoted x the quotient space under is the quotient set y equipped with the quotient topology, that is the topology whose open sets are the subsets u. This may sound somewhat abstract, and indeed, it is. With this in mind, in figure 1 we have a diagram of how one might do this with v r2 and w x,y. Exercise and solution manual for a first course in linear algebra. The first isomorphism theorem of linear algebra says that the quotient space v ker t is isomorphic to the image of v in w.
It is the totality of all the vectors after the transformation. Linear algebra abridged is generated from linear algebra done right by sheldon axler, third edition by excluding all proofs, examples, and exercises, along with most comments. The author writes the book that may be confusing to the first time student of linear algebra. Then modding out by u amounts to zeroing out the components of the basis corresponding to u, i. Applications version 1 by howard anton and chris rorres and linear algebra and its applications 10 by gilbert strang are loaded with applications. Advanced linear algebra 1st edition hugo woerdeman. Please be aware, however, that the handbook might contain. A vector space quotient is a very simple projection when viewed in an appropriate basis. Xi the linear map induced by a on the quotient space xi. Kazdan topics 1 basics 2 linear equations 3 linear maps 4 rank one matrices 5 algebra of matrices 6 eigenvalues and eigenvectors 7 inner products and quadratic forms 8 norms and metrics 9 projections and re. Vectors, linear equations, matrix algebra, determinants, eigenvalues and eigenvectors, linear transformations, dimension, similarity and diagonalizability, complex numbers, projection theorem, gramschmidt orthonormalization, qr factorization, least squares approximation, orthogonal.
A first course in linear algebra lyryx learning inc. Preface here are my online notes for my linear algebra course that i teach here at lamar university. Advanced linear algebra features a studentfriendly approach to the theory of linear algebra. Aug 15, 2016 how to think about linear systems of equations geometrically. The quotient set, y x is the set of equivalence classes of elements of x. So, if you are have studied the basic notions of abstract algebra, the concept of a coset will be familiar to you. It is assumed that the reader has already studied matrix algebra or linear algebra, however, these notes are completely selfcontained. It is assumed that the reader is already familiar with this subject, if not with the exact terminology and notation used here. Quotient spaces oklahoma state universitystillwater. F the set of polynomials with degree 6 n, whose coe. Another important space associated with the matrix is the row space. Linear algebracolumn and row spaces wikibooks, open books. Nov 04, 2010 they fill up all of w like the pages of a closed book.
However, even if you have not studied abstract algebra, the idea of a coset in a vector space is very natural. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space. If you are to use this book, i suggest to also pick up a copy of another linear algebra book that may concentrate on definitions and simpler proofs in the begining of the book. We will describe a construction of the quotient vector space vu. This handbook is intended to assist graduate students with qualifying examination preparation. It is used by the pure mathematician and by the mathematically trained scientists of all disciplines. The applied viewpoint taken here is motivated by the study of mechanical systems and electrical networks, in which the notation and methods of linear algebra play an important role. Dual spaces friday 3 november 2005 lectures for part a of oxford fhs in mathematics and joint schools linear functionals and the dual space dual bases annihilators an example the second dual.
Linear algebra is one of the most applicable areas of mathematics. Such things form the subject matter of linear geometry. One can, but in general should not, try to visualize the quotient space v w as a subspace of the space v. Some simple results several observations should be made. Like its name suggests it is built entirely out of the rows of the matrix. In linear algebra, the quotient of a vector space v by a subspace n is a vector space obtained by collapsing n to zero.
Introductiontovectorspaces,vector algebras,andvectorgeometries richard a. A set with a linear structure in the sense of our discussion is called a linear space or vector space. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. If you are a student and nd the level at which many of the current beginning linear algebra. While this may contradict the experience of many experienced mathematicians, the approach here is consciously algebraic. Many elementary proofs are omitted, but the experienced reader will have no di. The full version of linear algebra done right is available at and in both printed and electronic forms. Such vectors belong to the foundation vector space rn of all vector spaces.
A first course in linear algebra presents an introduction to the fascinating subject of linear algebra for students who have a reasonable understanding of basic algebra. An immediate corollary, for finitedimensional spaces, is the ranknullity theorem. He focuses on finite fields and complex numbers, and discusses matrix algebra over. These notes are for a graduate course in linear algebra. Any vector space is an abelian group under the operation of vector addition. First, we generalize the lie algebraic structure of general linear algebra gl n, r to this dimensionfree quotient space. The algebra of linear transformation, ranknullity theorem, matrix and linear transformation, rank and nullity of a matrix, isomorphism.
Controlled and conditioned invariants in linear system theory. X be a topological space, and let be an equivalence relation on x. Those readers who have studied abstract algebra can relate the null space to the kernel of a homomorphism and the column space to the range. The elements of the new vector space vw are the cosets. The collection of all the pages is vw and becomes a new vector space, with the addition and scalar multiplication defined in your pdf file. Smith october 14, 2011 abstract an introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative.
Despite the fact that these are my class notes they should be accessible to anyone wanting to learn linear algebra or needing a refresher. Linear algebra the subject of linear algebra includes the solution of linear equations, a topic properly belonging to college algebra. My main goal in writing these notes was to give to the student a concise overview of the main concepts,ideas and results that usually are covered in the rst course on. Linear algebraquotient space wikibooks, open books for an. Inverse matrices, column space and null space essence of. The material is developed completely from scratch, but at a faster pace than a beginning linear algebra course. These notes do assume that the reader has a good working knowledge of basic algebra. U vis isomorphic to u xdx cy the coset yin xand x is called a coset representative for x xy the quotient space module y p n. The geometric aspect of linear algebra involves lines, planes, and their higher dimensional analogues. This is an incredibly useful notion, which we will use from time to time to simplify other tasks. Exercise and solution manual for a first course in linear algebra robert a. Solutions of nonhomogeneous system writing solution set in parametric vector form jiwen he, university of houston math 2331, linear algebra 2 12.